Chapter 5: Problem 30
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 1 } ^ { 4 } - x ^ { - 2 } d x$$
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In Exercises \(49-54,\) use NINT to solve the problem. For what value of \(x\) does \(\int_{0}^{x} e^{-t^{2}} d t=0.6 ?\)
Extending the ldeas Writing to Learn If \(f\) is an odd continuous function, give a graphical argument to explain why \(\int_{0}^{x} f(t) d t\) is even.
Rectangular Approximation Methods Show that if \(f\) is a nonnegative function on the interval \([a, b]\) and the line \(x=(a+b) / 2\) is a line of symmetry of the graph of \(y=f(x)\) then \(L R A M_{n} f=\operatorname{RRAM}_{n} f\) for every positive integer \(n .\)
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.) $$\int_{0}^{\pi} \sin x d x$$
.In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{0}^{\pi} \sin x d x$$
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