Studdy AI Logo
Math  /  Calculus

Question16. Express ∭Exz3dV\iiint_{E} x z^{3} d V, where E={(x,y,z):0≤x≤1,0≤y≤x,0≤z≤x2y2}E=\left\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq x, 0 \leq z \leq x^{2} y^{2}\right\} in a form that can be evaluated. A. ∫0x2y2∫0x∫01xz3dxdydz\int_{0}^{x^{2} y^{2}} \int_{0}^{x} \int_{0}^{1} x z^{3} d x d y d z B. ∫0x∫01∫0x2y2xz2dzdxdy\int_{0}^{x} \int_{0}^{1} \int_{0}^{x^{2} y^{2}} x z^{2} d z d x d y C. ∫01∫0x2y2∫0xxz2dydzdx\int_{0}^{1} \int_{0}^{x^{2} y^{2}} \int_{0}^{x} x z^{2} d y d z d x D. ∫01∫0x∫0x2y2xz2dzdydx\int_{0}^{1} \int_{0}^{x} \int_{0}^{x^{2} y^{2}} x z^{2} d z d y d x

Studdy Solution

STEP 1

1. The region E E is defined by the inequalities 0≤x≤1 0 \leq x \leq 1 , 0≤y≤x 0 \leq y \leq x , and 0≤z≤x2y2 0 \leq z \leq x^2 y^2 .
2. We need to express the given triple integral in a form that matches one of the provided options.
3. The order of integration must respect the limits of the region E E .

STEP 2

1. Analyze the region E E and determine the correct order of integration.
2. Match the limits of integration with one of the provided options.

STEP 3

Analyze the region E E defined by the inequalities:
- 0≤x≤1 0 \leq x \leq 1 - 0≤y≤x 0 \leq y \leq x - 0≤z≤x2y2 0 \leq z \leq x^2 y^2
This suggests that x x is the outermost variable, followed by y y , and then z z as the innermost variable.

STEP 4

Express the triple integral in the form:
∫01∫0x∫0x2y2xz3 dz dy dx \int_{0}^{1} \int_{0}^{x} \int_{0}^{x^2 y^2} x z^3 \, dz \, dy \, dx
This matches option D, where the integration order is dz dy dx dz \, dy \, dx and the limits correspond to the region E E .
The correct expression is:
∫01∫0x∫0x2y2xz3 dz dy dx \boxed{\int_{0}^{1} \int_{0}^{x} \int_{0}^{x^2 y^2} x z^3 \, dz \, dy \, dx}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord