Calculate the double integral of f(x, y) = 100 - 6x²y dA for the region R: 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.

Question image

Understand the Problem

The question is asking us to calculate a double integral of the function f(x, y) = 100 - 6x²y over the specified region R, which is defined by the constraints for x and y. We will need to set up and evaluate the integral accordingly.

Answer

The result of the double integral is \( 400 \).
Answer for screen readers

The final value of the double integral is

$$ 400 $$

Steps to Solve

  1. Set up the double integral

The double integral we need to evaluate is given by:

$$ \iint_R (100 - 6x^2y) , dA $$

with the region ( R ) defined by ( 0 \leq x \leq 2 ) and ( -1 \leq y \leq 1 ).

  1. Iterate the integral

We will integrate first with respect to ( y ) and then with respect to ( x ):

$$ \int_{0}^{2} \left( \int_{-1}^{1} (100 - 6x^2y) , dy \right) dx $$

  1. Evaluate the inner integral

Calculate the inner integral:

$$ \int_{-1}^{1} (100 - 6x^2y) , dy $$

This will involve finding the integral of each term separately.

$$ = \int_{-1}^{1} 100 , dy - \int_{-1}^{1} 6x^2y , dy $$

  1. Compute the first part

Calculating the first integral:

$$ \int_{-1}^{1} 100 , dy = 100[y]_{-1}^{1} = 100(1 - (-1)) = 100(2) = 200 $$

  1. Compute the second part

Now for the second integral:

$$ \int_{-1}^{1} 6x^2y , dy = 6x^2 \left[ \frac{y^2}{2} \right]_{-1}^{1} = 6x^2 \left( \frac{(1)^2}{2} - \frac{(-1)^2}{2} \right) = 6x^2(0) = 0 $$

Thus, the inner integral becomes:

$$ \int_{-1}^{1} (100 - 6x^2y) , dy = 200 - 0 = 200 $$

  1. Evaluate the outer integral

Now substitute into the outer integral:

$$ \int_{0}^{2} 200 , dx $$

Calculating this integral gives:

$$ 200[x]_{0}^{2} = 200(2 - 0) = 400 $$

The final value of the double integral is

$$ 400 $$

More Information

The double integral represents the volume under the surface defined by ( f(x, y) = 100 - 6x^2y ) over the rectangular region ( R ).

Tips

  • Forgetting to handle bounds correctly can lead to incorrect evaluations, especially when integrating over symmetric regions.
  • Miscalculating the integral of polynomials like ( 6x^2y ) can lead to mistakes in the final result.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser