Chapter 4

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{2} x^{2} d x ; \quad n=4 $$

Let \(F(x)=\int_{2}^{x} t^{2} d t\). a. Use Part 1 of the Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). b. Use Part 2 of the Fundamental Theorem of Calculus to integrate \(\int_{2}^{x} t^{2} d t\) to obtain an alternative expression for \(F(x) .\) c. Differentiate the expression for \(F(x)\) found in part (b), and compare the result with that obtained in part (a). Comment on your result.

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=x, \quad[0,1], \quad n= $$

In Exencises \(1-6\), find the integral using the indicated substitution. $$ \int(2 x+3)^{5} d x, \quad u=2 x+3 $$

Find the indefinite integral, and check your answer by differentiation. $$ \int(x+2) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{1}^{3}\left(x^{2}-1\right) d x ; \quad n=6 $$

Repeat Exercise 1 with \(G(x)=\int_{0}^{x} \sqrt{3 t+1} d t\).

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=x, \quad[1,4], \quad n= $$

Find the integral using the indicated substitution. $$ \int x^{2} \sqrt{x^{3}+2} d x, \quad u=x^{3}+2 $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(6 x^{2}-2 x+1\right) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{1}^{2} x^{3} d x, \quad n=6 $$

Find the derivative of the function. $$ F(x)=\int_{0}^{x} \sqrt{3 t+5} d t $$

You are given a function \(f\) defined on an interval \([a, b]\), the number \(n\) of subintervals of equal length \(\Delta x=(b-a) / n\), and the evaluation points \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right] .\) (a) Sketch the graph of \(f\) and the rectangles associated with the Riemann sum for f on \([a, b]\), and \((\) b) find the Riemann sum. \(f(x)=2 x-3, \quad[0,2], \quad n=4, \quad c_{k}\) is the midpoint

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=2 x+3, \quad[0,4], \quad n=5, \quad c_{k} \text { is the right endpoint } $$

Find the integral using the indicated substitution. $$ \int \frac{x}{\sqrt{x^{2}+1}} d x, \quad u=x^{2}+1 $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(3-2 x+x^{2}\right) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{1}^{2} \frac{1}{x^{2}} d x ; \quad n=4 $$

Find the derivative of the function. $$ G(x)=\int_{-1}^{x} t \sqrt{t^{2}+1} d t $$

You are given a function \(f\) defined on an interval \([a, b]\), the number \(n\) of subintervals of equal length \(\Delta x=(b-a) / n\), and the evaluation points \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right] .\) (a) Sketch the graph of \(f\) and the rectangles associated with the Riemann sum for f on \([a, b]\), and \((\) b) find the Riemann sum. \(f(x)=-2 x+1, \quad[-1,2], \quad n=6, \quad c_{k}\) is the left endpoint

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=3-2 x, \quad[0,1], \quad n=5, \quad c_{k} \text { is the left endpoint } $$

Find the integral using the indicated substitution. $$ \int e^{-3 x} d x, \quad u=-3 x $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(x^{3}-2 x^{2}+x+1\right) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{2} x \sqrt{2 x^{2}+1} d x ; \quad n=6 $$

Find the derivative of the function. $$ g(x)=\int_{2}^{x} \frac{1}{t^{2}+1} d t $$

You are given a function \(f\) defined on an interval \([a, b]\), the number \(n\) of subintervals of equal length \(\Delta x=(b-a) / n\), and the evaluation points \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right] .\) (a) Sketch the graph of \(f\) and the rectangles associated with the Riemann sum for f on \([a, b]\), and \((\) b) find the Riemann sum. \(f(x)=\sqrt{x}-1, \quad[0,3], \quad n=6, \quad c_{k}\) is the right endpoint

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=8-2 x, \quad[1,3], \quad n=4, \quad c_{k} \text { is the midpoint } $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(2 x^{9}+3 e^{x}+4\right) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{1} e^{-x} d x ; \quad n=6 $$

Find the derivative of the function. $$ h(x)=\int_{x}^{3} \frac{t}{\sqrt{t+1}} d t $$

You are given a function \(f\) defined on an interval \([a, b]\), the number \(n\) of subintervals of equal length \(\Delta x=(b-a) / n\), and the evaluation points \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right] .\) (a) Sketch the graph of \(f\) and the rectangles associated with the Riemann sum for f on \([a, b]\), and \((\) b) find the Riemann sum. \(f(x)=2 \sin x,\left[0, \frac{5 \pi}{4}\right], \quad n=5, \quad c_{k}\) is the right endpoint

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=x^{2}, \quad[0,1], \quad n=5, \quad c_{k} \text { is the right endpoint } $$

Find the integral using the indicated substitution. $$ \int \frac{\sin x}{\cos ^{2} x} d x, \quad u=\cos x $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(2 x^{2 / 3}-4 x^{1 / 3}+4\right) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{1} x e^{=x^{2}} d x ; \quad n=6 $$

Find the derivative of the function. $$ F(x)=\int_{x}^{\pi} \sin 2 t d t $$

Use Equation (2) to evaluate the integral. \(\int_{0}^{2} x d x\)

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=x^{2}, \quad[1,3], \quad n=4, \quad c_{k} \text { is the midpoint } $$

In Exercises \(7-72\), find the indefinite integral. $$ \int 2 x\left(x^{2}+1\right)^{4} d x $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right) d x $$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{\pi / 2} \cos 2 x d x ; \quad n=6 $$

Find the derivative of the function. $$ G(x)=\int_{0}^{x^{2}} t \sin t d t $$

Use Equation (2) to evaluate the integral. \(\int_{-1}^{2} x^{2} d x\)

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=4-x^{2}, \quad[0,2], \quad n=8, \quad c_{k} \text { is the left endpoint } $$

Find the indefinite integral. $$ \int x^{2}\left(2 x^{3}-1\right)^{4} d x $$

Find the indefinite integral, and check your answer by differentiation. $$ \int x^{2 / 3}(x-1) d x $$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $$ \int_{0}^{1} \frac{d x}{2 x+1} ; n=7 $$

Find the derivative of the function. $$ g(x)=\int_{2}^{\sqrt{x}} \frac{\sin t}{t} d t $$

Use Equation (2) to evaluate the integral. \(\int_{-1}^{3}(x-2) d x\)

Find the indefinite integral. $$ \int(2 x-4)^{3 / 5} d x $$

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(e^{t}+t^{e}\right) d t $$