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Math  /  Calculus

QuestionEvaluate the definite integral ∫0π10sin⁡(x)dx\int_{0}^{\pi} 10 \sin (x) d x

Studdy Solution

STEP 1

1. We are given the definite integral ∫0π10sin⁡(x) dx\int_{0}^{\pi} 10 \sin(x) \, dx.
2. We need to evaluate this integral over the interval from 00 to π\pi.

STEP 2

1. Identify the integral to be solved.
2. Determine the antiderivative of the integrand.
3. Evaluate the antiderivative at the upper and lower limits of integration.
4. Subtract the evaluated lower limit from the evaluated upper limit to find the definite integral.

STEP 3

Identify the integral to be solved:
The integral is ∫0π10sin⁡(x) dx\int_{0}^{\pi} 10 \sin(x) \, dx.

STEP 4

Determine the antiderivative of the integrand:
The integrand is 10sin⁡(x)10 \sin(x). The antiderivative of sin⁡(x)\sin(x) is −cos⁡(x)-\cos(x). Therefore, the antiderivative of 10sin⁡(x)10 \sin(x) is:
−10cos⁡(x) -10 \cos(x)

STEP 5

Evaluate the antiderivative at the upper and lower limits of integration:
Evaluate at the upper limit π\pi:
−10cos⁡(π)=−10(−1)=10 -10 \cos(\pi) = -10(-1) = 10
Evaluate at the lower limit 00:
−10cos⁡(0)=−10(1)=−10 -10 \cos(0) = -10(1) = -10

STEP 6

Subtract the evaluated lower limit from the evaluated upper limit:
10−(−10)=10+10=20 10 - (-10) = 10 + 10 = 20
The value of the definite integral is:
20 \boxed{20}

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