Problem 7
Question
Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{-1}^{0}(x-2) d x $$
Step-by-Step Solution
Verified Answer
The value of the definite integral is -1.
1Step 1: Find the antiderivative
Find an antiderivative \(F(x)\) of \(f(x) = x - 2\). An antiderivative of \(x\) is \(\frac{1}{2}x^2\) and an antiderivative of \(-2\) is \(-2x\). So, \(F(x) = \frac{1}{2}x^2 - 2x\).
2Step 2: Evaluate the antiderivative at the limits of integration
Evaluate \(F(x)\) at 0 and -1. Given \(F(x) = \frac{1}{2}x^2 - 2x\), we find \(F(0) = \frac{1}{2}(0)^2 - 2(0) = 0\) and \(F(-1) = \frac{1}{2}(-1)^2 - 2(-1) = -1 + 2 = 1\).
3Step 3: Calculate the definite integral
Compute \(F(b) - F(a)\) to get the value of the definite integral. In this case, that is equal to \(F(0) - F(-1) = 0 - 1 = -1\).
Other exercises in this chapter
Problem 7
In Exercises \(7-12,\) verify the identity. \(\tanh ^{2} x+\operatorname{sech}^{2} x=1\)
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Find the integral. $$ \int \frac{1}{\sqrt{1-(x+1)^{2}}} d x $$
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In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{1}^{2}\left(x^{2}+1\right) d x $$
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Find the indefinite integral and check the result by differentiation. $$ \int(1+2 x)^{4}(2) d x $$
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