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Wize University Calculus 1 Textbook > Applications of Integration

Area Between Curves

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Area Between Curves

Definite integrals give us the exact area under the curve of a graph. What about if we wanted to find the area between curves? The integral of the difference of two functions answers this question.



Area Between Two Curves

If f(x)≥g(x)f(x)\geq g(x) for xx in [a,b][a,b], then the area above g(x)g(x) and below f(x)f(x) between x=ax=a and x=bx=b is given by

∫ab[f(x)−g(x)]dx\boxed{\int_a^b\left[f(x)-g(x)\right]dx}


Watch Out!
If the functions intersect multiple times, it may result in multiple regions that are bounded between the two curves.


We need to find the point of intersection between the curves, then we may need to break up the integral into multiple parts to ensure each area added up is positive.

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Example: Area Between Curves

Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=0x=0 and x=5x=5.




Intersection
2x2+10=4x+162x2−4x−6=0(2x−6)(x+1)=0x=3,−1\begin{aligned} &2x^2+10=4x+16\\ &2x^2-4x-6=0\\ &(2x-6)(x+1)=0\\ &x=3,-1 \end{aligned}

Area Between The Curves

∫03[(4x+16)−(2x2+10)]dx+∫35[(2x2+10)−(4x+16)]dx\displaystyle \int_0^3[(4x+16)-(2x^2+10)]dx+\int_3^5[(2x^2+10)-(4x+16)]dx

∫03(−2x2+4x+6)dx+∫35(2x2−4x−6)dx\displaystyle \int_0^3(-2x^2+4x+6)dx+\int_3^5(2x^2-4x-6)dx

=(−2x33+2x2+6x)∣03+(2x33−2x2−6x)∣35\displaystyle =\left(\frac{-2x^3}{3}+2x^2+6x\right)\bigg|_0^3+\left(\frac{2x^3}{3}-2x^2-6x\right)\bigg|_3^5

=(−2(3)33+2(3)2+6(3))−(−2(0)33+2(0)2+6(0))+(2(5)33−2(5)2−6(5))−(2(3)33−2(3)2−6(3))\displaystyle =\left(\frac{-2(3)^3}{3}+2(3)^2+6(3)\right)\displaystyle -\left(\frac{-2(0)^3}{3}+2(0)^2+6(0)\right)+\left(\frac{2(5)^3}{3}-2(5)^2-6(5)\right)-\left(\frac{2(3)^3}{3}-2(3)^2-6(3)\right)

=(−18+18+18)+(2503−50−30)−(18−18−18)\displaystyle =\left(-18+18+18\right)\displaystyle +\left(\frac{250}{3}-50-30\right)-\left(18-18-18\right)

=18+(2503−80)+18\displaystyle =18+\left( \frac{250}{3}-80\right)+18

=36+(2503−2403)\displaystyle = 36+\left(\frac{250}{3}-\frac{240}{3}\right)

=36+103≈39.33=\displaystyle 36+\frac{10}{3} \approx39.33
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Example: Area Between Curves

Find the area of the shaded region below


Rearranging for x: x=2−y2  and  x=2−yx=2-y^2\ \text{ and }\ x=2-y

Point of Intersection:
2−y2=2−y2-y^2=2-y
0=2−y−2+y20=2-y-2+y^2
0=y2−y0=y^2-y
0=y(y−1)0=y\left(y-1\right)
y=0,1y=0,1

Based on the graph, x=2−y2x=2-y^2 is the rightmost curve and x=2−yx=2-yis the leftmost curve.

A=∫01[(2−y2)−(2−y)]dy\displaystyle A=\int_0^1\left[\left(2-y^2\right)-\left(2-y\right)\right]dy

=∫01[2−y2−2+y]dy\displaystyle =\int_0^1\left[2-y^2-2+y\right]dy

=∫01[−y2+y]dy\displaystyle =\int_0^1\left[-y^2+y\right]dy

=[−y33+y22]01\displaystyle =\left[-\frac{y^3}{3}+\frac{y^2}{2}\right]_0^1

=(−13+12)−0\displaystyle =\left(-\frac{1}{3}+\frac{1}{2}\right)-0

=−26+36=16\displaystyle =-\frac{2}{6}+\frac{3}{6}=\boxed{\frac{1}{6}}

Practice: Area Between Curves

Find the area bounded by the curves y=x2−x  and  y=3x−3 y=x^2-x\ \text{ and }\ y=3x-3\


Extra Practice
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=x2−x and y=3x−3 y=x^2-x\ and\ y=3x-3\
Area between curves
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Area between curves
Find the area of the region bounded between x2−x+1x^2-x+1 and the line y=1y=1.
Area Between Curves
Find the area of the shaded region below:
Area Between Curves
Given the functions y=x3y=x^3 and y=2x−x2y=2x-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⁡xy=\cos x and y=sin⁡xy=\sin x between x=0x=0 and x=π4x=\dfrac{\pi}{4}.
Find the area of the region bounded by y=4x2−2x+1y=4x^2-2x+1 between x=−2x=-2 and x=4x=4 and the x-axis.
Find the area of the region bounded by y=∣x∣  and  y=x2−6.y=|x|\ \text{ and }\ y=x^2-6.
Find the area bounded by the curves y=ex,y=x,x=−1  and  x=3y=e^x,y=x,x=-1\ \text{ and }\ x=3.
Area between curves
Find the area of the region bounded between x2−x+1x^2-x+1 and the line y=1y=1.
Area Between Curves
Find the area of the region bounded by the curves y=x3−1y=x^3-1 and y=x−1y=x-1
Area Between Curves
Find the area of the region bounded by the curves y=x3−1y=x^3-1 and y=x−1y=x-1
Area between curves
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Area between curves
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Area between curves
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Area between curves
Find the area of the region bounded between x2−x+1x^2-x+1 and the line y=1y=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=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Area Between Curves
Find integral that represents the area enclosed by the curves y=x3y=x^3 and y=xy=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=1xy=\dfrac{1}{\sqrt{x}} , 𝑦=10𝑦 = 10 and 𝑥=1𝑥 = 1, using vertical slices. Do not solve.
Practice: Horizontal Slices
Q:\textbf{Q:} Find the area of the region bounded by the curves 𝑥=𝑦2−1𝑥 = 𝑦^2 − 1 and 𝑦=−𝑥+5𝑦 = −𝑥 + 5.
Horizontal and Vertical Slices
Write an integral that represents the area of the region bounded by y=x−4y=\sqrt{x-4}, 𝑦=2𝑦 = 2, 𝑥=10𝑥 = 10, using first vertical slices, then horizontal slices. Do not solve.
Area Between The Curves: Area in Pieces
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Area Between Curves: Parabola and Horizontal Line
Find the area of the region bounded by the curve y=x2−x+1y=x^2-x+1 and the line y=1y=1.
area btw curves
Find the area bounded by the curves y=x2−x and y=3x−3 y=x^2-x\ and\ y=3x-3\
Find the area enclosed by the curves 𝑦 = 𝑥2 + 1 and 𝑦 = 3𝑥 − 1.
(a) 66 (b) 16\frac16 (c) 107\frac{10}{7} (d) 17\frac17 (e) 11
Find the area enclosed by the curves 𝑦 = 𝑥2 + 1 and 𝑦 = 3𝑥 − 1.
Find the area bounded by the curves y=ex,y=x,x=−1  and  x=3y=e^x,y=x,x=-1\ \text{ and }\ x=3.
Find the area bounded by y=cos⁡xy=\cos x and y=sin⁡xy=\sin x between x=0x=0 and x=π4x=\dfrac{\pi}{4}.
Find the area of the region bounded by y=4x2−2x+1y=4x^2-2x+1 between x=−2x=-2 and x=4x=4 and the x-axis.
Find the area of the region bounded by y=ln⁡ xx and  y=(ln⁡ x)2x.y=\frac{\ln\ x}{x} \text{ and }\ y=\frac{\left(\ln\ x\right)^2}{x}.
Find the area of the region bounded by y=∣x∣  and  y=x2−6.y=|x|\ \text{ and }\ y=x^2-6.
Find the area bounded by the curves y=ex,y=x,x=−1  and  x=3y=e^x,y=x,x=-1\ \text{ and }\ x=3.
Practice: Area between Curves
Find the area of the region bounded by the curves y=x3−1y=x^3-1 and y=x−1y=x-1
Practice: Area between Curves
Find the integral that represents the area of the region bounded by the curevs x=y2−4y+2x=y^2-4y+2 and 5−x=y2−y5-x=y^2-y.
Area Between Curves
Given the functions y=x3y=x^3 and y=2x−x2y=2x-x^2,
a) Sketch the region enclosed by these graphs
b) Find the area of the enclosed region from part a)
Area between curves
Find the area of the region bounded between x2−x+1x^2-x+1 and the line y=1y=1.
Area between curves
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Find the area bounded by the curves y=x2−x and y=3x−3 y=x^2-x\ and\ y=3x-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=exy=e^x, x=1x=1, andy=3y=3?
i. ∫1ln⁡3ex−3dx\int_1^{\ln3}e^x-3dx
ii. ∫e3ln⁡y−1dy\int_e^3\ln y-1dy
Area between curves
Find the area bounded by the curves y=x2−x and y=3x−3 y=x^2-x\ and\ y=3x-3\
Find the area under the function f(x)=cosxf(x)=cosx from x=−π2x=-\frac{\pi}{2} to x=π2x=\frac{\pi}{2}
Which of the following represents the area between y=exy=e^x , y=8y=8 and the y-axis?
Which of the following represents the area bounded by the curves y=cos⁡xy=\cos x, y=2cos⁡xy=2\cos x, x=0x=0, and x=πx=\pi.
Area Between Areas
Find the area between the functions y=x2−4x+4y=x^2-4x+4 and y=4−x2y=4-x^2.
Find the area of the shaded region below
Area Between Curves
Find the area enclosed by y=xy=\sqrt{x} and y=x2y=x^2.
Find the area enclosed by the curves 𝑦 = −𝑥 + 2 and 𝑥 + 𝑦2 = 2.
What is the area enclosed between f(x)=x4+x2f\left(x\right)=x^4+x^2 and g(x)=x4+xg\left(x\right)=x^4+x in the first quadrant?
Area Between Curves
Find the area of the region bounded by the curves y=x3−1y=x^3-1 and y=x−1y=x-1
Practice: Area between Curves
Find the integral that represents the area of the region bounded by the curevs x=y2−4y+2x=y^2-4y+2 and 5−x=y2−y5-x=y^2-y.
Area Between Curves
Find the area enclosed by the curves y=x−4,  y=1,  x=13y=\sqrt{x-4},\ \ y=1,\ \ x=13.
Extra Practice:
Try doing this using 2 different methods (one with the equations set up as y=...y=... and the other with the equations set up as x=...x=...)
Find the area enclosed by x=1+yx=1+\sqrt{y} and x=(2+y)/2x=(2+y)/2.
Find the area of the region bounded by y=4x2−2x+1y=4x^2-2x+1 between x=−2x=-2 and x=4x=4 and the x-axis.
Find the area of the region bounded by y=ln⁡ xx and  y=(ln⁡ x)2x.y=\frac{\ln\ x}{x} \text{ and }\ y=\frac{\left(\ln\ x\right)^2}{x}.
Find the area bounded by the curves y=ex,y=x,x=−1  and  x=3y=e^x,y=x,x=-1\ \text{ and }\ x=3.
Vertical Slices
Write the integral that represents the area of the region bounded by y=1xy=\dfrac{1}{\sqrt{x}} , 𝑦=10𝑦 = 10 and 𝑥=1𝑥 = 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−4y=\sqrt{x-4}, 𝑦=2𝑦 = 2, 𝑥=10𝑥 = 10, using first vertical slices, then horizontal slices. Do not solve.
Practice: Horizontal Slices
Q:\textbf{Q:} Find the area of the region bounded by the curves 𝑥=𝑦2−1𝑥 = 𝑦^2 − 1 and 𝑦=−𝑥+5𝑦 = −𝑥 + 5.
Area between curves
What is the area enclosed between f(x)=x4+x2f\left(x\right)=x^4+x^2 and g(x)=x4+xg\left(x\right)=x^4+x in the first quadrant?
Find the area bounded by y=cos⁡xy=\cos x and y=sin⁡xy=\sin x between x=0x=0 and x=π4x=\dfrac{\pi}{4}.
Practice: Area in Pieces
Q.\textbf{Q.} Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−3x=-3 and x=5x=5.
Area Between Curves: Parabola and Horizontal Line
Find the area of the region bounded by the curve y=x2−x+1y=x^2-x+1 and the line y=1y=1.
Area Between The Curves: Area in Pieces
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
Write the integral that represents the area under the curve 𝑦 = 𝑥2 + 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=x2−6.y=|x|\ \text{ and }\ y=x^2-6.
Find the area enclosed by y=sin⁡(πx/2)y=\sin(\pi x/2) and y=xy=x.
Find the area enclosed by y=x3−2xy=x^3-2x and y=2xy=2x.
Find the area enclosed by y=1/xy=1/x, y=1/x2y=1/x^2, and x=3x=3.
Find the area enclosed by the curves 𝑦 = 𝑥2 + 1 and 𝑦 = 3𝑥 − 1.
Which of the following integral would compute the area of the bounded region enclosed by the parabola y=x2−3y=x^2-3 and the line y=5x+3 ?y=5x+3\ ?
Find the area bounded by the curves y=x2−x  and  y=3x−3 y=x^2-x\ \text{ and }\ y=3x-3\ .
Find the area enclosed by y=2x2+10y=2x^2+10 and y=4x+16y=4x+16 between x=−2x=-2 and x=5x=5.
area btw curves
Find the area bounded by the curves y=x2−x and y=3x−3 y=x^2-x\ and\ y=3x-3\
Area between curves
Which of the following represents the area bounded by the curves y=cos⁡xy=\cos x, y=2cos⁡xy=2\cos x, x=0x=0, and x=πx=\pi.
Area Between Curves
Which of the following represent(s) the area of the region bounded by y=exy=e^x, x=1x=1, andy=3y=3?
i. ∫1ln⁡3ex−3dx\displaystyle \int_1^{\ln3}e^x-3dx
ii. ∫e3ln⁡y−1dy\displaystyle \int_e^3\ln y-1dy
Area Between Curves
Find integral that represents the area enclosed by the curves y=x3y=x^3 and y=xy=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=x2+1 and  y=3x−1y=x^2+1\ and\ \ y=3x-1.