Calculate the double integral of f(x, y) = 100 - 6x²y dA for the region R: 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.
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
- 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 ).
- Iterate the integral
We will integrate first with respect to ( y ) and then with respect to ( x ):
$$ \int_{0}^{2} \left( \int_{-1}^{1} (100 - 6x^2y) , dy \right) dx $$
- 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.
$$ = \int_{-1}^{1} 100 , dy - \int_{-1}^{1} 6x^2y , dy $$
- Compute the first part
Calculating the first integral:
$$ \int_{-1}^{1} 100 , dy = 100[y]_{-1}^{1} = 100(1 - (-1)) = 100(2) = 200 $$
- 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:
$$ \int_{-1}^{1} (100 - 6x^2y) , dy = 200 - 0 = 200 $$
- 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.
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