Problem 16
In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{1}^{2} \frac{1}{x} d x$$
Problem 17
In Exercises \(1-20,\) find \(d y / d x\).. $$y=\int_{\sqrt{x}}^{0} \sin \left(r^{2}\right) d r$$
Problem 17
In Exercises \(15-18,\) find the average value of the function on the interval without integrating, by appealing to the geometry of the region between the graph and the \(x\) -axis. $$f ( t ) = \sin t , \quad [ 0,2 \pi ]$$
Problem 17
In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{4} \sqrt{x} d x$$
Problem 18
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{3 \sqrt{2}}^{10} \ln \left(2+p^{2}\right) d p$$
Problem 18
In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{\pi} \sin x d x$$
Problem 18
In Exercises \(15-18,\) find the average value of the function on the interval without integrating, by appealing to the geometry of the region between the graph and the \(x\) -axis. $$f ( \theta ) = \tan \theta , \quad \left[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right]$$
Problem 19
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) \(\int_{\pi}^{2 \pi} \sin x d x\)
Problem 19
Consider the integral \(\int_{-1}^{3}\left(x^{3}-2 x\right) d x\) (a) Use Simpson's Rule with \(n=4\) to approximate its value. (b) Find the exact value of the integral. What is the error, \(\left|E_{S}\right| ?\) (c) Explain how you could have predicted what you found in (b) from knowing the error-bound formula. (d) Writing to Learn Is it possible to make a general statement about using Simpson's Rule to approximate integrals of cubic polynomials? Explain.
Problem 20
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { \cos x } ^ { \pi / 2 } \cos x d x$$
