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Find 6 ff 3 xy dydx. Write your answer in exact for

Find 6 ff 3 xy dydx. Write your answer in exact for

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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3.2 5
**Problem:** Find the double integral \( \int_{3}^{4} \int_{3}^{6} xy \, dy \, dx \). Write your answer in exact form. --- **Solution:** To solve this problem, we need to evaluate the given double integral, which models the volume under the surface \( z = xy \) over the specified region in the \( xy \)-plane. **Steps:** 1. **Evaluate the Inner Integral:** The inner integral is with respect to \( y \), from 3 to 6: \[ \int_{3}^{6} xy \, dy \] Treat \( x \) as a constant: \[ = x \int_{3}^{6} y \, dy \] The antiderivative of \( y \) is \( \frac{y^2}{2} \), so: \[ = x \left[ \frac{y^2}{2} \right]_{3}^{6} \] \[ = x \left( \frac{6^2}{2} - \frac{3^2}{2} \right) \] \[ = x \left( \frac{36}{2} - \frac{9}{2} \right) \] \[ = x \left( 18 - 4.5 \right) \] \[ = x \times 13.5 \] \[ = 13.5x \] 2. **Evaluate the Outer Integral:** Now integrate with respect to \( x \), from 3 to 4: \[ \int_{3}^{4} 13.5x \, dx \] The antiderivative of \( 13.5x \) is \( \frac{13.5x^2}{2} \), so: \[ = \left[ \frac{13.5x^2}{2} \right]_{3}^{4} \] \[ = \frac{13.5}{2} \left( 4^2 - 3^2 \right) \] \[ = \frac{13.5}{2} \
Transcribed Image Text:**Problem:** Find the double integral \( \int_{3}^{4} \int_{3}^{6} xy \, dy \, dx \). Write your answer in exact form. --- **Solution:** To solve this problem, we need to evaluate the given double integral, which models the volume under the surface \( z = xy \) over the specified region in the \( xy \)-plane. **Steps:** 1. **Evaluate the Inner Integral:** The inner integral is with respect to \( y \), from 3 to 6: \[ \int_{3}^{6} xy \, dy \] Treat \( x \) as a constant: \[ = x \int_{3}^{6} y \, dy \] The antiderivative of \( y \) is \( \frac{y^2}{2} \), so: \[ = x \left[ \frac{y^2}{2} \right]_{3}^{6} \] \[ = x \left( \frac{6^2}{2} - \frac{3^2}{2} \right) \] \[ = x \left( \frac{36}{2} - \frac{9}{2} \right) \] \[ = x \left( 18 - 4.5 \right) \] \[ = x \times 13.5 \] \[ = 13.5x \] 2. **Evaluate the Outer Integral:** Now integrate with respect to \( x \), from 3 to 4: \[ \int_{3}^{4} 13.5x \, dx \] The antiderivative of \( 13.5x \) is \( \frac{13.5x^2}{2} \), so: \[ = \left[ \frac{13.5x^2}{2} \right]_{3}^{4} \] \[ = \frac{13.5}{2} \left( 4^2 - 3^2 \right) \] \[ = \frac{13.5}{2} \
Expert Solution
Step 1: Define the problem

The given integral integral subscript 3 superscript 4 integral subscript 3 superscript 6 x y d y d x

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