Check the solution to "Compute the volume to z=y2,z=1,2x+z=4,x=0"

geduiwelh

Answered question

2021-09-17

Compute the volume to

z = y^{2}, z = 1, 2 x + z = 4, x = 0

Mayme

Skilled2021-09-18Added 103 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

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

z = y^{2}

to find the limits of y. Also, find the limits of x.

1 = y^{2}

y = \pm 1

y=-1,1
2x+z=4

x = 2 - \frac{z}{2}

Step 2
Now, solve the x and z integrals.

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

= \int_{- 1}^{1} \int_{y^{2}}^{1} (2 - \frac{z}{2}) d z d y

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

= \int_{- 1}^{1} [(2 - \frac{1}{4}) - (2 y^{2} - \frac{y^{4}}{4})] d y

= \int_{- 1}^{1} (\frac{7}{4} - 2 y^{2} + \frac{y^{4}}{4}) d y

Evaluate the y integral to find the volume.

V = \int_{- 1}^{1} (\frac{7}{4} - 2 y^{2} + \frac{y^{4}}{4}) d y

= (\frac{7 y}{4} - \frac{2 y^{3}}{3} + \frac{y^{5}}{20}) ∣_{- 1}^{1}

= 2 (\frac{7}{4} - \frac{2}{3} + \frac{1}{20})

= \frac{34}{15}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?