Problem 58
Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b] .\) To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=x^{3}+x ; a=0, b=1 $$
Problem 58
Decide whether the given statement is true or false. Then justify your answer. If \(f(x) \geq 0\) and \(\int_{a}^{b} f(x) d x=0\), then \(f(x)=0\) for all \(x\) in \([a, b]\).
Problem 59
Decide whether the given statement is true or false. Then justify your answer. $$\text { If } \begin{aligned} \int_{a}^{b} f(x) d x &>\int_{a}^{b} g(x) d x, \text { then } \\ \int_{a}^{b}[f(x)-g(x)] d x>0 \end{aligned}$$
Problem 59
, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x $$
Problem 60
, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-1}^{1} x^{2} \cosh x^{3} d x $$
Problem 60
Decide whether the given statement is true or false. Then justify your answer. If \(f\) and \(g\) are continuous and \(f(x)>g(x)\) for all \(x\) in \([a, b]\), then \(\left|\int_{a}^{b} f(x) d x\right|>\left|\int_{a}^{b} g(x) d x\right| .\)
Problem 61
The velocity of an object is \(v(t)=2-|t-2|\). Assuming that the object is at the origin at time 0, find a formula for its position at time \(t\).
Problem 61
Let \(A_{a}^{b}\) denote the area under the curve \(y=x^{2}\) over the interval \([a, b]\). (a) Prove that \(A_{0}^{b}=b^{3} / 3 .\) Hint \(: \Delta x=b / n\), so \(x_{i}=i b / n ;\) use circumscribed polygons. (b) Show that \(A_{a}^{b}=b^{3} / 3-a^{3} / 3\). Assume that \(a \geq 0\).
Problem 62
Suppose that an object, moving along the \(x\) -axis, has velocity \(v=t^{2}\) meters per second at time \(t\) seconds. How far did it travel between \(t=3\) and \(t=5\) ? See Problem 61 .
Problem 62
The velocity of an object is
$$v(t)=\left\\{\begin{array}{ll}
5 & \text { if } 0 \leq t \leq 100 \\
6-t / 100 & \text { if } 100
