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Area Between Curves
Related Topics
Wize University Calculus 1 Textbook > Applications of Integration
Area Between Curves
4 Activities
Given the functions
y
=
x
3
y=x^3
y
=
x
3
and
y
=
2
x
−
x
2
y=2x-x^2
y
=
2
x
−
x
2
,
a) Sketch the region enclosed by these graphs
b) Find the area of the enclosed region from part a)
Part b) Area =
I don't know
Check Submission
More Area Between Curves Questions:
Area Between Curves
Find the area of the shaded region below
Area Between Curves
Practice: Area Between Curves
Find the area of the shaded region below
Area between curves
Find the area bounded by the curves
y
=
x
2
−
x
a
n
d
y
=
3
x
−
3
y=x^2-x\ and\ y=3x-3\
y
=
x
2
−
x
an
d
y
=
3
x
−
3
Area between curves
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Area between curves
Find the area of the region bounded between
x
2
−
x
+
1
x^2-x+1
x
2
−
x
+
1
and the line
y
=
1
y=1
y
=
1
.
Area Between Curves
Find the area of the shaded region below:
Area Between Curves
Given the functions
y
=
x
3
y=x^3
y
=
x
3
and
y
=
2
x
−
x
2
y=2x-x^2
y
=
2
x
−
x
2
,
a) Sketch the region enclosed by these graphs
b) Find the area of the enclosed region from part a)
Find the area bounded by
y
=
cos
x
y=\cos x
y
=
cos
x
and
y
=
sin
x
y=\sin x
y
=
sin
x
between
x
=
0
x=0
x
=
0
and
x
=
π
4
x=\dfrac{\pi}{4}
x
=
4
π
.
Find the area of the region bounded by
y
=
4
x
2
−
2
x
+
1
y=4x^2-2x+1
y
=
4
x
2
−
2
x
+
1
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
4
x=4
x
=
4
and the
x
-axis.
Find the area of the region bounded by
y
=
∣
x
∣
and
y
=
x
2
−
6.
y=|x|\ \text{ and }\ y=x^2-6.
y
=
∣
x
∣
and
y
=
x
2
−
6.
Find the area bounded by the curves
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
y=e^x,y=x,x=-1\ \text{ and }\ x=3
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
.
Area between curves
Find the area of the region bounded between
x
2
−
x
+
1
x^2-x+1
x
2
−
x
+
1
and the line
y
=
1
y=1
y
=
1
.
Area Between Curves
Find the area of the region bounded by the curves
y
=
x
3
−
1
y=x^3-1
y
=
x
3
−
1
and
y
=
x
−
1
y=x-1
y
=
x
−
1
Area Between Curves
Find the area of the region bounded by the curves
y
=
x
3
−
1
y=x^3-1
y
=
x
3
−
1
and
y
=
x
−
1
y=x-1
y
=
x
−
1
Area between curves
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Area between curves
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Area between curves
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Area between curves
Find the area of the region bounded between
x
2
−
x
+
1
x^2-x+1
x
2
−
x
+
1
and the line
y
=
1
y=1
y
=
1
.
Area Between Curves
Find the area of the shaded region below
Area Between The Curves: Area in Pieces
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Area Between Curves
Find integral that represents the area enclosed by the curves
y
=
x
3
y=x^3
y
=
x
3
and
y
=
x
y=x
y
=
x
.
Area Between Curves
Practice: Area Between Curves
Find the area of the shaded region below
Vertical Slices
Write the integral that represents the area of the region bounded by
y
=
1
x
y=\dfrac{1}{\sqrt{x}}
y
=
x
1
,
𝑦
=
10
𝑦 = 10
y
=
10
and
𝑥
=
1
𝑥 = 1
x
=
1
, using vertical slices. Do not solve.
Practice: Horizontal Slices
Q:
\textbf{Q:}
Q:
Find the area of the region bounded by the curves
𝑥
=
𝑦
2
−
1
𝑥 = 𝑦^2 − 1
x
=
y
2
−
1
and
𝑦
=
−
𝑥
+
5
𝑦 = −𝑥 + 5
y
=
−
x
+
5
.
Horizontal and Vertical Slices
Write an integral that represents the area of the region bounded by
y
=
x
−
4
y=\sqrt{x-4}
y
=
x
−
4
,
𝑦
=
2
𝑦 = 2
y
=
2
,
𝑥
=
10
𝑥 = 10
x
=
10
, using first vertical slices, then horizontal slices. Do not solve.
Area Between The Curves: Area in Pieces
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Area Between Curves: Parabola and Horizontal Line
Find the area of the region bounded by the curve
y
=
x
2
−
x
+
1
y=x^2-x+1
y
=
x
2
−
x
+
1
and the line
y
=
1
y=1
y
=
1
.
area btw curves
Find the area bounded by the curves
y
=
x
2
−
x
a
n
d
y
=
3
x
−
3
y=x^2-x\ and\ y=3x-3\
y
=
x
2
−
x
an
d
y
=
3
x
−
3
Find the area enclosed by the curves 𝑦 = 𝑥
2
+ 1 and 𝑦 = 3𝑥 − 1.
(a)
6
6
6
(b)
1
6
\frac16
6
1
(c)
10
7
\frac{10}{7}
7
10
(d)
1
7
\frac17
7
1
(e)
1
1
1
Find the area enclosed by the curves 𝑦 = 𝑥
2
+ 1 and 𝑦 = 3𝑥 − 1.
Find the area bounded by the curves
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
y=e^x,y=x,x=-1\ \text{ and }\ x=3
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
.
Find the area bounded by
y
=
cos
x
y=\cos x
y
=
cos
x
and
y
=
sin
x
y=\sin x
y
=
sin
x
between
x
=
0
x=0
x
=
0
and
x
=
π
4
x=\dfrac{\pi}{4}
x
=
4
π
.
Find the area of the region bounded by
y
=
4
x
2
−
2
x
+
1
y=4x^2-2x+1
y
=
4
x
2
−
2
x
+
1
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
4
x=4
x
=
4
and the
x
-axis.
Find the area of the region bounded by
y
=
ln
x
x
and
y
=
(
ln
x
)
2
x
.
y=\frac{\ln\ x}{x} \text{ and }\ y=\frac{\left(\ln\ x\right)^2}{x}.
y
=
x
l
n
x
and
y
=
x
(
l
n
x
)
2
.
Find the area of the region bounded by
y
=
∣
x
∣
and
y
=
x
2
−
6.
y=|x|\ \text{ and }\ y=x^2-6.
y
=
∣
x
∣
and
y
=
x
2
−
6.
Find the area bounded by the curves
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
y=e^x,y=x,x=-1\ \text{ and }\ x=3
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
.
Practice: Area between Curves
Find the area of the region bounded by the curves
y
=
x
3
−
1
y=x^3-1
y
=
x
3
−
1
and
y
=
x
−
1
y=x-1
y
=
x
−
1
Practice: Area between Curves
Find the integral that represents the area of the region bounded by the curevs
x
=
y
2
−
4
y
+
2
x=y^2-4y+2
x
=
y
2
−
4
y
+
2
and
5
−
x
=
y
2
−
y
5-x=y^2-y
5
−
x
=
y
2
−
y
.
Area between curves
Find the area of the region bounded between
x
2
−
x
+
1
x^2-x+1
x
2
−
x
+
1
and the line
y
=
1
y=1
y
=
1
.
Area between curves
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Find the area bounded by the curves
y
=
x
2
−
x
a
n
d
y
=
3
x
−
3
y=x^2-x\ and\ y=3x-3\
y
=
x
2
−
x
an
d
y
=
3
x
−
3
Practice: Integration
Practice: Area Between Curves
Find the area of the shaded region below
Which of the following represent the area of the region bounded by
y
=
e
x
y=e^x
y
=
e
x
,
x
=
1
x=1
x
=
1
, and
y
=
3
y=3
y
=
3
?
i.
∫
1
ln
3
e
x
−
3
d
x
\int_1^{\ln3}e^x-3dx
∫
1
l
n
3
e
x
−
3
d
x
ii.
∫
e
3
ln
y
−
1
d
y
\int_e^3\ln y-1dy
∫
e
3
ln
y
−
1
d
y
Area between curves
Find the area bounded by the curves
y
=
x
2
−
x
a
n
d
y
=
3
x
−
3
y=x^2-x\ and\ y=3x-3\
y
=
x
2
−
x
an
d
y
=
3
x
−
3
Find the area under the function
f
(
x
)
=
c
o
s
x
f(x)=cosx
f
(
x
)
=
cos
x
from
x
=
−
π
2
x=-\frac{\pi}{2}
x
=
−
2
π
to
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
Which of the following represents the area between
y
=
e
x
y=e^x
y
=
e
x
,
y
=
8
y=8
y
=
8
and the y-axis?
Which of the following represents the area bounded by the curves
y
=
cos
x
y=\cos x
y
=
cos
x
,
y
=
2
cos
x
y=2\cos x
y
=
2
cos
x
,
x
=
0
x=0
x
=
0
, and
x
=
π
x=\pi
x
=
π
.
Area Between Areas
Find the area between the functions
y
=
x
2
−
4
x
+
4
y=x^2-4x+4
y
=
x
2
−
4
x
+
4
and
y
=
4
−
x
2
y=4-x^2
y
=
4
−
x
2
.
Find the area of the shaded region below
Area Between Curves
Find the area enclosed by
y
=
x
y=\sqrt{x}
y
=
x
and
y
=
x
2
y=x^2
y
=
x
2
.
Find the area enclosed by the curves 𝑦 = −𝑥 + 2 and 𝑥 + 𝑦
2
= 2.
What is the area enclosed between
f
(
x
)
=
x
4
+
x
2
f\left(x\right)=x^4+x^2
f
(
x
)
=
x
4
+
x
2
and
g
(
x
)
=
x
4
+
x
g\left(x\right)=x^4+x
g
(
x
)
=
x
4
+
x
in the first quadrant?
Area Between Curves
Find the area of the region bounded by the curves
y
=
x
3
−
1
y=x^3-1
y
=
x
3
−
1
and
y
=
x
−
1
y=x-1
y
=
x
−
1
Practice: Area between Curves
Find the integral that represents the area of the region bounded by the curevs
x
=
y
2
−
4
y
+
2
x=y^2-4y+2
x
=
y
2
−
4
y
+
2
and
5
−
x
=
y
2
−
y
5-x=y^2-y
5
−
x
=
y
2
−
y
.
Area Between Curves
Find the area enclosed by the curves
y
=
x
−
4
,
y
=
1
,
x
=
13
y=\sqrt{x-4},\ \ y=1,\ \ x=13
y
=
x
−
4
,
y
=
1
,
x
=
13
.
Extra Practice:
Try doing this using 2 different methods (one with the equations set up as
y
=
.
.
.
y=...
y
=
...
and the other with the equations set up as
x
=
.
.
.
x=...
x
=
...
)
Find the area enclosed by
x
=
1
+
y
x=1+\sqrt{y}
x
=
1
+
y
and
x
=
(
2
+
y
)
/
2
x=(2+y)/2
x
=
(
2
+
y
)
/2
.
Find the area of the region bounded by
y
=
4
x
2
−
2
x
+
1
y=4x^2-2x+1
y
=
4
x
2
−
2
x
+
1
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
4
x=4
x
=
4
and the
x
-axis.
Find the area of the region bounded by
y
=
ln
x
x
and
y
=
(
ln
x
)
2
x
.
y=\frac{\ln\ x}{x} \text{ and }\ y=\frac{\left(\ln\ x\right)^2}{x}.
y
=
x
l
n
x
and
y
=
x
(
l
n
x
)
2
.
Find the area bounded by the curves
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
y=e^x,y=x,x=-1\ \text{ and }\ x=3
y
=
e
x
,
y
=
x
,
x
=
−
1
and
x
=
3
.
Vertical Slices
Write the integral that represents the area of the region bounded by
y
=
1
x
y=\dfrac{1}{\sqrt{x}}
y
=
x
1
,
𝑦
=
10
𝑦 = 10
y
=
10
and
𝑥
=
1
𝑥 = 1
x
=
1
, using vertical slices. Do not solve.
Horizontal and Vertical Slices
Write an integral that represents the area of the region bounded by
y
=
x
−
4
y=\sqrt{x-4}
y
=
x
−
4
,
𝑦
=
2
𝑦 = 2
y
=
2
,
𝑥
=
10
𝑥 = 10
x
=
10
, using first vertical slices, then horizontal slices. Do not solve.
Practice: Horizontal Slices
Q:
\textbf{Q:}
Q:
Find the area of the region bounded by the curves
𝑥
=
𝑦
2
−
1
𝑥 = 𝑦^2 − 1
x
=
y
2
−
1
and
𝑦
=
−
𝑥
+
5
𝑦 = −𝑥 + 5
y
=
−
x
+
5
.
Area between curves
What is the area enclosed between
f
(
x
)
=
x
4
+
x
2
f\left(x\right)=x^4+x^2
f
(
x
)
=
x
4
+
x
2
and
g
(
x
)
=
x
4
+
x
g\left(x\right)=x^4+x
g
(
x
)
=
x
4
+
x
in the first quadrant?
Find the area bounded by
y
=
cos
x
y=\cos x
y
=
cos
x
and
y
=
sin
x
y=\sin x
y
=
sin
x
between
x
=
0
x=0
x
=
0
and
x
=
π
4
x=\dfrac{\pi}{4}
x
=
4
π
.
Practice: Area in Pieces
Q.
\textbf{Q.}
Q.
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
3
x=-3
x
=
−
3
and
x
=
5
x=5
x
=
5
.
Area Between Curves: Parabola and Horizontal Line
Find the area of the region bounded by the curve
y
=
x
2
−
x
+
1
y=x^2-x+1
y
=
x
2
−
x
+
1
and the line
y
=
1
y=1
y
=
1
.
Area Between The Curves: Area in Pieces
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
Write the integral that represents the area under the curve 𝑦 = 𝑥
2
+ 3 and above the
x
-axis on the interval [−1,3]. Do not solve.
Find the area of the region bounded by
y
=
∣
x
∣
and
y
=
x
2
−
6.
y=|x|\ \text{ and }\ y=x^2-6.
y
=
∣
x
∣
and
y
=
x
2
−
6.
Find the area enclosed by
y
=
sin
(
π
x
/
2
)
y=\sin(\pi x/2)
y
=
sin
(
π
x
/2
)
and
y
=
x
y=x
y
=
x
.
Find the area enclosed by
y
=
x
3
−
2
x
y=x^3-2x
y
=
x
3
−
2
x
and
y
=
2
x
y=2x
y
=
2
x
.
Find the area enclosed by
y
=
1
/
x
y=1/x
y
=
1/
x
,
y
=
1
/
x
2
y=1/x^2
y
=
1/
x
2
, and
x
=
3
x=3
x
=
3
.
Find the area enclosed by the curves 𝑦 = 𝑥
2
+ 1 and 𝑦 = 3𝑥 − 1.
Which of the following integral would compute the area of the bounded region enclosed by the parabola
y
=
x
2
−
3
y=x^2-3
y
=
x
2
−
3
and the line
y
=
5
x
+
3
?
y=5x+3\ ?
y
=
5
x
+
3
?
Find the area bounded by the curves
y
=
x
2
−
x
and
y
=
3
x
−
3
y=x^2-x\ \text{ and }\ y=3x-3\
y
=
x
2
−
x
and
y
=
3
x
−
3
.
Find the area enclosed by
y
=
2
x
2
+
10
y=2x^2+10
y
=
2
x
2
+
10
and
y
=
4
x
+
16
y=4x+16
y
=
4
x
+
16
between
x
=
−
2
x=-2
x
=
−
2
and
x
=
5
x=5
x
=
5
.
area btw curves
Find the area bounded by the curves
y
=
x
2
−
x
a
n
d
y
=
3
x
−
3
y=x^2-x\ and\ y=3x-3\
y
=
x
2
−
x
an
d
y
=
3
x
−
3
Area between curves
Which of the following represents the area bounded by the curves
y
=
cos
x
y=\cos x
y
=
cos
x
,
y
=
2
cos
x
y=2\cos x
y
=
2
cos
x
,
x
=
0
x=0
x
=
0
, and
x
=
π
x=\pi
x
=
π
.
Area Between Curves
Which of the following represent(s) the area of the region bounded by
y
=
e
x
y=e^x
y
=
e
x
,
x
=
1
x=1
x
=
1
, and
y
=
3
y=3
y
=
3
?
i.
∫
1
ln
3
e
x
−
3
d
x
\displaystyle \int_1^{\ln3}e^x-3dx
∫
1
l
n
3
e
x
−
3
d
x
ii.
∫
e
3
ln
y
−
1
d
y
\displaystyle \int_e^3\ln y-1dy
∫
e
3
ln
y
−
1
d
y
Area Between Curves
Find integral that represents the area enclosed by the curves
y
=
x
3
y=x^3
y
=
x
3
and
y
=
x
y=x
y
=
x
.
Area Between Curves
Practice: Area Between Curves
Find the area of the shaded region below
Area between curves
Find the area enclosed by the curves
y
=
x
2
+
1
a
n
d
y
=
3
x
−
1
y=x^2+1\ and\ \ y=3x-1
y
=
x
2
+
1
an
d
y
=
3
x
−
1
.