91Ó°ÊÓ

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?

The linear function \(f(x)=3-x\) is decreasing on the interval \([0,3] .\) Is the area function for \(f\) (with left endpoint 0 ) increasing or decreasing on the interval [0,3]\(?\) Draw a picture and explain.

When using a change of variables \(u=g(x)\) to evaluate the definite integral \(\int_{a}^{b} f(g(x)) g^{\prime}(x) d x,\) how are the limits of integration transformed?

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 \(f(x)=x^{2}\).

Evaluate \(\int_{0}^{2} 3 x^{2} d x\) and \(\int_{-2}^{2} 3 x^{2} d x\)

Sketch the function \(y=x\) on the interval [0,2] and let \(R\) be the region bounded by \(y=x\) and the \(x\) -axis on \([0,2] .\) Now sketch a rectangle in the first quadrant whose base is [0,2] and whose area equals the area of \(R\).

If the change of variables \(u=x^{2}-4\) is used to evaluate the definite integral \(\int_{2}^{4} f(x) d x,\) what are the new limits of integration?

Explain in words and express mathematically the inverse relationship between differentiation and integration as given by Part 1 of the Fundamental Theorem of Calculus.

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

Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval \([a, b] ?\) Explain.