Question
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
Answers
GPT 3.5
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.
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.