Chapter 4

Use the given substitution to evaluate the indicated integral. $$\int x^{2} \sqrt{x^{3}+2} d x, u=x^{3}+2$$

Compute Midpoint, Trapezoidal and Simpson's Rule approximations by hand (leave your answer as a fraction) for \(n=4\). $$\int_{0}^{1}\left(x^{2}+1\right) d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{2}(2 x-3) d x$$

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy). $$\int_{0}^{3}\left(x^{3}+x\right) d x$$

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. $$f(x)=x^{2}+1, \quad \text { (a) }[0,1], n=4 ; \quad \text { (b) }[0,2], n=4$$

A calculation is described in words. Translate each into summation notation and then compute the sum. The sum of the squares of the first 50 positive integers.

In exercises \(1-4,\) sketch several members of the family of functions defined by the antiderivative. $$\int x^{3} d x$$

Use the given substitution to evaluate the indicated integral. $$\int x^{3}\left(x^{4}+1\right)^{-2 / 3} d x, u=x^{4}+1$$

Compute Midpoint, Trapezoidal and Simpson's Rule approximations by hand (leave your answer as a fraction) for \(n=4\). $$\int_{0}^{2}\left(x^{2}+1\right) d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{3}\left(x^{2}-2\right) d x$$

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy). $$\int_{0}^{3} \sqrt{x^{2}+1} d x$$

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. $$f(x)=x^{3}-1, \quad \text { (a) }[1,2], n=4 ; \quad \text { (b) }[1,3], n=4$$

A calculation is described in words. Translate each into summation notation and then compute the sum. The square of the sum of the first 50 positive integers.

Sketch several members of the family of functions defined by the antiderivative. $$\int\left(x^{3}-x\right) d x$$

Use the given substitution to evaluate the indicated integral. $$\int \frac{(\sqrt{x}+2)^{3}}{\sqrt{x}} d x, u=\sqrt{x}+2$$

Compute Midpoint, Trapezoidal and Simpson's Rule approximations by hand (leave your answer as a fraction) for \(n=4\). $$\int_{1}^{3} \frac{1}{x} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{-1}^{1}\left(x^{3}+2 x\right) d x$$

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy). $$\int_{0}^{\pi} \sin x^{2} d x$$

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. \(f(x)=\sin x\) (a) \([0, \pi], n=4\) (b) \([0, \pi], n=8\)

A calculation is described in words. Translate each into summation notation and then compute the sum. The sum of the square roots of the first 10 positive integers.

Sketch several members of the family of functions defined by the antiderivative. $$\int\left(x^{3}-x\right) d x$$

Use the given substitution to evaluate the indicated integral. $$\int \sin x \cos x \, d x, u=\sin x$$

Compute Midpoint, Trapezoidal and Simpson's Rule approximations by hand (leave your answer as a fraction) for \(n=4\). $$\int_{-1}^{1}\left(2 x-x^{2}\right) d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{2}\left(x^{3}+3 x-1\right) d x$$

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy). $$\int_{-2}^{2} e^{-x^{2}} d x$$

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. $$f(x)=4-x^{2}, \quad \text { (a) }[-1,1], n=4 ; \quad \text { (b) }[-3,-1], n=4$$

Sketch several members of the family of functions defined by the antiderivative. $$\int \cos x \, d x$$

Evaluate the indicated integral. $$\int x^{3} \sqrt{x^{4}+3} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{4}(\sqrt{x}+3 x) d x$$

Evaluate the integral by computing the limit of Riemann sums. $$\int_{0}^{1} 2 x d x$$

Approximate the area under the curve on the given interval using \(n\) rectangles and the evaluation rules (a) left endpoint (b) midpoint (e) right endpoint. $$y=x^{2}+1 \text { on }[0,1], n=16$$

Write out all terms and compute the sums. $$\sum_{i=1}^{6} 3 i^{2}$$

Find the general antiderivative. $$\int\left(3 x^{4}-3 x\right) d x$$

Evaluate the indicated integral. $$\int \sec ^{2} x \sqrt{\tan x} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{1}^{2}\left(4 x-2 / x^{2}\right) d x$$

Evaluate the integral by computing the limit of Riemann sums. $$\int_{1}^{2} 2 x d x$$

Approximate the area under the curve on the given interval using \(n\) rectangles and the evaluation rules (a) left endpoint (b) midpoint (e) right endpoint. $$y=x^{2}+1 \text { on }[0,2], n=16$$

Write out all terms and compute the sums. $$\sum_{i=3}^{7}\left(i^{2}+i\right)$$

Find the general antiderivative. $$\int\left(x^{3}-2\right) d x$$

Evaluate the indicated integral. $$\begin{aligned} &4\\\ &\int \frac{\sin x}{\sqrt{\cos x}} d x \end{aligned}$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{1}\left(x \sqrt{x}+x^{1 / 3}\right) d x$$

Evaluate the integral by computing the limit of Riemann sums. $$\int_{0}^{2} x^{2} d x$$

Approximate the area under the curve on the given interval using \(n\) rectangles and the evaluation rules (a) left endpoint (b) midpoint (e) right endpoint. $$y=\sqrt{x+2} \text { on }[1,4], n=16$$

Write out all terms and compute the sums. $$\sum_{i=6}^{10}(4 i+2)$$

Find the general antiderivative. $$\int\left(3 \sqrt{x}-\frac{1}{x^{4}}\right) d x$$

Evaluate the indicated integral. $$\int \sin ^{3} x \cos x d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{8}\left(\sqrt[3]{x}-x^{2 / 3}\right) d x$$

Evaluate the integral by computing the limit of Riemann sums. $$\int_{0}^{3}\left(x^{2}+1\right) d x$$

Approximate the area under the curve on the given interval using \(n\) rectangles and the evaluation rules (a) left endpoint (b) midpoint (e) right endpoint. $$y=e^{-2 x} \text { on }[-1,1], n=16$$

Write out all terms and compute the sums. $$\sum_{i=6}^{8}\left(i^{2}+2\right)$$