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Chapter 5: Problem 38

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{4} x(x-2)(x-4) d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the function $$f(x) = x(x-2)(x-4)$$ over the interval $$[0, 4]$$. Answer: The value of the definite integral is $$52$$.

Step by step solution

01

Find the antiderivative of the function

To find the antiderivative of $$f(x) = x(x-2)(x-4)$$, we first need to expand the polynomial using the distributive property. We get: $$f(x) = x(x-2)(x-4) = x(x^2 - 6x + 8) = x^3 - 6x^2 + 8x$$ Now, to find the antiderivative $$F(x)$$, we will perform term-by-term integration: $$F(x) = \int(x^3 - 6x^2 + 4x) d x = \frac{1}{4}x^4 - 2x^3 + 4x^2 + C$$ However, since we are dealing with a definite integral, we don't need the constant of integration, so: $$F(x) = \frac{1}{4}x^4 - 2x^3 + 4x^2$$
02

Apply the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus, we will substitute the values of our interval (0 and 4) into the antiderivative function and find the difference: $$\int_{0}^{4} f(x) d x = F(4) - F(0)$$ We'll start by finding $$F(4)$$: $$F(4) = \frac{1}{4}(4)^4 - 2(4)^3 + 4(4)^2 = 16 - 128 + 64 = 52$$ And now we'll find $$F(0)$$: $$F(0) = \frac{1}{4}(0)^4 - 2(0)^3 + 4(0)^2 = 0$$ Finally, we'll find the difference: $$\int_{0}^{4} f(x) d x = F(4) - F(0) = 52 - 0 = 52$$ So the value of the definite integral is $$52$$.

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Most popular questions from this chapter

Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \cos (\pi \sqrt{t}) d t$$

What value of \(b>-1\) maximizes the integral $$\int_{-1}^{b} x^{2}(3-x) d x ?$$

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\frac{1}{x} ; a=1, b=4, c=6$$

Use geometry and the result of Exercise 76 to evaluate the following integrals. $$\int_{1}^{6} f(x) d x, \text { where } f(x)=\left\\{\begin{array}{ll}2 x & \text { if } 1 \leq x<4 \\\10-2 x & \text { if } 4 \leq x \leq 6\end{array}\right.$$

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=e^{x} ; a=0, b=\ln 2, c=\ln 4$$
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