## Exercise 133

1 Evaluate the following:

2 Evaluate the following:

3 In Fig. 13.1a, take the lowest value of the function attained in each subinterval as the height of the rectangular block, i.e., take f(x2) instead of f(xt) as the height of the first block, though still retaining Ax, as its width, and do likewise for the other blocks.

(a) Write a summation expression for the total area A** of the new rectangles.

(b) Does A** overestimate or underestimate the desired area A?

(c) Would A** tend to approach or to deviate further from A if a finer segmentation of [a, b] were introduced? (Hint: Try a diagram.)

(d) In the limit, when the number n of subintervals approaches oo, would the approximation value A** approach the true value A, just as the approximation value A* did?

(<?) What can you conclude from the above about the Riemann integrability of the function /(x) in the figure?

4 The definite integral f f(x) dx is said to represent an area under a curve. Does this

Ja curve refer to the graph of the integrand f(x), or of the primitive function F(x)7 If we plot the graph of the F(x) function, how can we show the above definite integral on itâ€”by an area, a line segment, or a point?

5 Verify that a constant c can be equivalently expressed as a definite integral:

rb c rc

0 0