91Ó°ÊÓ

Evaluate \(\int_{3}^{8} f^{\prime}(t) d t,\) where \(f^{\prime}\) is continuous on \([3,8], f(3)=4\) and \(f(8)=20\)

Suppose the interval [1,3] is partitioned into \(n=4\) subintervals. What is the subinterval length \(\Delta x\) ? List the grid points \(x_{0}, x_{1}, x_{2}\) \(x_{3},\) and \(x_{4} .\) Which points are used for the left, right, and midpoint Riemann sums?

Use a substitution of the form \(u=a x+b\) to evaluate the following indefinite integrals. $$\int(x+1)^{12} d x$$

Explain how the notation for Riemann sums, \(\sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x_{k},\) corresponds to the notation for the definite integral, \(\int_{a}^{b} f(x) d x\)

Evaluate \(\int_{2}^{7} 3 d x\) using the Fundamental Theorem of Calculus. Check your work by evaluating the integral using geometry.

Use a substitution of the form \(u=a x+b\) to evaluate the following indefinite integrals. $$\int e^{3 x+1} d x$$

Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-200}^{200} 2 x^{5} d x$$

Suppose the interval [2,6] is partitioned into \(n=4\) subintervals with grid points \(x_{0}=2, x_{1}=3, x_{2}=4, x_{3}=5,\) and \(x_{4}=6\) Write, but do not evaluate, the left, right, and midpoint Riemann sums for \(f(x)=x^{2}\).

Use a substitution of the form \(u=a x+b\) to evaluate the following indefinite integrals. $$\int \sqrt{2 x+1} d x$$

Give a geometrical explanation of why \(\int_{a}^{a} f(x) d x=0\)