91Ó°ÊÓ

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=2-x^{3}, y=2+x^{3}, 0 \leq x \leq 1\)

Use the definition of the Riemann integral in terms of Riemann sums to prove property (3) of definite integrals. That is, if \(f(x)\) is continuous on \([a, b]\) and \(k\) is any constant, then: $$ \int^{b} k f(x) d x=k \int^{b} f(x) d x $$

Compute the indefinite integrals. $$ \int\left(x^{7 / 2}+x^{2 / 7}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=e^{x}, y=e^{-x}, 0 \leq x \leq 2\)

Use a diagram to explain why, if \(f(x)\) is continuous on an interval that contains all of the points \(a, b, c\), then $$ \int_{a}^{b} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x $$ That is, derive property (5) of definite integrals. You should consider the cases (a) \(b

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=\sqrt{1-x^{2}}, y=1,-1 \leq x \leq 1\)

Given that \(\int_{0}^{u} x^{2} d x=\frac{1}{3} a^{3}\) evaluate the following: (a) \(\int_{0}^{1} \frac{1}{2} x^{2} d x\) (b) \(\int_{0}^{-1} 3 x^{2} d x\) (c) \(\int_{-1}^{2} \frac{1}{3} x^{2} d x\) (d) \(\int_{1}^{1} 3 x^{2} d x\) (e) \(\int_{-2}^{3}(x+1)^{2} d x\) (f) \(\int_{2}^{4}(x-2)^{2} d x\)

Compute the indefinite integrals. $$ \int\left(x^{3 / 5}+x^{5 / 3}\right) d x $$

Compute the indefinite integrals. $$ \int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=x, y=1,0 \leq x \leq 1\)