Question: "Compute the volume to z=x2,z=1,y=0 and y=2"

Harlen Pritchard

Answered question

2021-09-23

Compute the volume to
$z = x^{2}, z = 1, y = 0$ and $y = 2$

Ezra Herbert

Skilled2021-09-24Added 99 answers

Step 1
The volume is computed by the following formula.

V = \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} \int_{z_{1}}^{z_{2}} d z d y d x

Frst, find the limits of integration. Put z=1 in

z = x^{2}

to find the limits of x.

1 = x^{2}

x = \pm 1

x=-1,1
Step 2
Now, solve the y and z integrals.

V = \int_{- 1}^{1} \int_{0}^{2} \int_{x^{2}}^{1} d z d y d x

= \int_{- 1}^{1} \int_{0}^{2} (z) ∣_{x^{2}}^{1} d y d x

= \int_{- 1}^{1} \int_{0}^{2} (1 - x^{2}) d y d x

= \int_{- 1}^{1} (1 - x^{2}) y ∣_{0}^{2} d x

= \int_{- 1}^{1} 2 (1 - x^{2}) d x

Evaluate the x integral to find the volume.

V = \int_{- 1}^{1} 2 (1 - x^{2}) d x

= 2 (x - \frac{x^{3}}{3}) ∣_{- 1}^{1}

= 2 [(1 - \frac{1}{3}) - (- 1 + \frac{1}{3})]

= \frac{8}{3}

Jeffrey Jordon

Expert2021-11-08Added 2605 answers

Answer is given below (on video)

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