Evaluate the definite integral ∫−5025−x2−−−−−−√dx by considering the area under the graph of a function. (1 point) Responses 25π 25 π 50π 50 π 5π4 5 π 4 25π4 25 π 4 625π4
To evaluate the definite integral, we need to find the antiderivative of the given function, and then evaluate it at the limits of integration.
The antiderivative of √(−x^2) is ∫ √(−x^2) dx = (1/2)∫√(−x^2) d(−x^2) = (1/2)∫√u du, where u = −x^2, du = −2xdx.
Now, we can use a trigonometric substitution by letting u = x^2, so du = 2xdx.
∫√(−x^2) dx = (1/2)∫√u du = (1/2)∫u^(1/2) du = (1/3)u^(3/2) + C = (1/3)(−x^2)^(3/2) + C.
To evaluate the definite integral from -5 to -25, we have:
∫(−5025−x^2)^(−−−−√) dx = (1/3)(−25^2)^(3/2) − (1/3)(−5^2)^(3/2).
Simplifying, we get:
∫(−5025−x^2)^(−−−−√) dx = (1/3)(−625)^(3/2) − (1/3)(25)^(3/2) = (1/3)(−625)^(3/2) − (1/3)(5^3) = (1/3)(−625)^(3/2) − (1/3)(125) = (1/3)(−625)^(3/2) − 125.
Therefore, the value of the definite integral ∫(−5025−x^2)^(−−−−√) dx is 25π, which is the same as 25π4.
So, the correct answer is 25π4.