Evaluate the integral. Give result in exact form. [tex]\int_{\frac{\pi}{2}}^{\frac{\pi}{4}}(11 \cos (x)-1) d x=[/tex]
Answer (2)
Find the antiderivative of the integrand: 11 sin ( x ) − x .
Evaluate the antiderivative at the upper limit: 11 sin ( 4 π ) − 4 π = 2 11 2 − 4 π .
Evaluate the antiderivative at the lower limit: 11 sin ( 2 π ) − 2 π = 11 − 2 π .
Subtract the value at the lower limit from the value at the upper limit to get the final answer: 2 11 2 + 4 π − 11 .
Explanation
Problem Analysis We are asked to evaluate the definite integral ∫ 2 π 4 π ( 11 cos ( x ) − 1 ) d x . This involves finding the antiderivative of the integrand and then evaluating it at the limits of integration.
Finding the Antiderivative First, we find the antiderivative of 11 cos ( x ) − 1 . The antiderivative of 11 cos ( x ) is 11 sin ( x ) , and the antiderivative of − 1 is − x . Therefore, the antiderivative of 11 cos ( x ) − 1 is 11 sin ( x ) − x .
Evaluating at Upper Limit Next, we evaluate the antiderivative at the upper limit of integration, which is 4 π : 11 sin ( 4 π ) − 4 π = 11 ( 2 2 ) − 4 π
Evaluating at Lower Limit Then, we evaluate the antiderivative at the lower limit of integration, which is 2 π : 11 sin ( 2 π ) − 2 π = 11 ( 1 ) − 2 π = 11 − 2 π
Subtracting the Values Now, we subtract the value of the antiderivative at the lower limit from the value at the upper limit: \begin{align*} \left(11\left(\frac{\sqrt{2}}{2}\right) - \frac{\pi}{4}\right) - \left(11 - \frac{\pi}{2}\right) &= \frac{11\sqrt{2}}{2} - \frac{\pi}{4} - 11 + \frac{\pi}{2} \ &= \frac{11\sqrt{2}}{2} + \frac{\pi}{4} - 11 \end{align*}
Final Result Therefore, the exact value of the definite integral is 2 11 2 + 4 π − 11 .
Examples
Definite integrals are used extensively in physics, such as calculating the displacement of an object given its velocity function, or determining the work done by a force over a certain distance. In engineering, they can be used to find the area under a curve, which can represent quantities like the total amount of material needed for a construction project or the total energy consumption over a period of time. Understanding how to evaluate definite integrals is crucial for solving real-world problems in these fields.
To evaluate the integral ∫ 2 π 4 π ( 11 cos ( x ) − 1 ) d x , we find its antiderivative, evaluate it at the limits, and subtract the results. The final answer is 2 11 2 + 4 π − 11 .
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