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Find the area enclosed by the curves 𝑦 = 𝑥2 + 1 and 𝑦 = 3𝑥 − 1. (a) 6 (b)…

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Wize University Calculus 1 Textbook > Applications of Integration

Area Between Curves

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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
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Given the functions y=x3y=x^3 and y=2x−x2y=2x-x^2,
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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.
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Find the area of the region bounded between x2−x+1x^2-x+1 and the line y=1y=1.
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Find the area of the region bounded by the curves y=x3−1y=x^3-1 and y=x−1y=x-1
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Find the area of the region bounded by the curves y=x3−1y=x^3-1 and y=x−1y=x-1
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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.
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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.
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.
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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.