Math Problem Statement

Evaluate \( \int_{0}^{3} \int_{1}^{2} x^2 \, dy \, dx \)

Solution

The integral to evaluate is:

0312x2dydx\int_{0}^{3} \int_{1}^{2} x^2 \, dy \, dx

To solve this, we proceed as follows:

  1. First, evaluate the inner integral with respect to yy, treating x2x^2 as a constant:

12x2dy=x2[y]12=x2(21)=x2\int_{1}^{2} x^2 \, dy = x^2 \cdot \left[ y \right]_{1}^{2} = x^2 \cdot (2 - 1) = x^2

So, the expression simplifies to:

03x2dx\int_{0}^{3} x^2 \, dx

  1. Now, evaluate the outer integral:

03x2dx=[x33]03=333033=273=9\int_{0}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{3} = \frac{3^3}{3} - \frac{0^3}{3} = \frac{27}{3} = 9

Thus, the result of the integral is:

0312x2dydx=9\int_{0}^{3} \int_{1}^{2} x^2 \, dy \, dx = 9

Would you like a breakdown of each step or have any questions?

Here are five related questions you might find useful:

  1. How would the answer change if the limits of xx were different?
  2. What happens if x2x^2 is replaced with another function, say x3x^3?
  3. How would you approach this problem if the integration limits for yy were functions of xx?
  4. How do we evaluate double integrals in polar coordinates?
  5. What techniques can simplify more complex iterated integrals?

Tip: Always check if the order of integration can simplify the problem before calculating.

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Math Problem Analysis

Mathematical Concepts

Double Integration
Iterated Integrals

Formulas

\( \int_{a}^{b} f(x) \, dx \)
Evaluating inner and outer integrals separately

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

College Calculus or Advanced High School