7.7, #16. The statement is true. y^{2} − 1 is concave-up on [0, 1] since the graph of (y^{2} − 1) is a parabola opening upwards. Thus the midpoint rule underestimates ∫_{0}^{1}(y^{2} − 1) dy.

7.7, #17. The statement is false. The trapezoid rule is exact whenever the graph of f(x) is a line.

7.7, #18. The statement is false. It can happen that LEFT(4)≥ ∫_{a}^{b} f(x) dx if f(x) is not monotonic on [a, b]. For example, let f be the saw-tooth function

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Then LEFT(2)= 0, whereas LEFT(4)=∫_{0}^{4} f(x) dx = 2.

7.7, #19. The statement is true. The magnitude of the error in using left or right sums depend on the magnitude of the first derivative.

7.8, #2. ∫_{1}^{b} \frac{x}{4 + x^{2}} dx = \frac{1}{2} ln(4+x^{2})|_{1}^{b} = (1/2)(ln(4+b^{2})−ln 5) goes to infinity as b → ∞.

Thus the improper integral diverges.

7.8, #3. ∫_{0}^{b} e^{−x}x dx = e^{−x}(−1 − x)|_{0}^{b} = e^{−b}(−1 − b) + 1. This has limit 1 as b → ∞

since e^{−b} goes to zero much faster than −1 − b goes to −∞. Thus the improper integral converges.

7.8, #4. ∫_{b}^{0} \frac{e^{x}}{1 + e^{x}} dx = ln(1 + e^{x})|_{b}^{0} = ln 2 − ln(1 + e^{b}). This has limit ln 2 as b → −∞.