Calculating areas and volumes

calculus

Homework

Section 7.1 (pp.359-361) - #5-12, 13, 14, 15, 17, #21-26

Section 7.2 (pp.368-369) - #5-18, 19, 20

Section 7.3 (pp.376-377) - #5-12, #13-18

7.2 6

import graph;
size(400,IgnoreAspect);
real f(real x) { return 5x;}
 
fill((1,0)--(1,f(1))--(2,f(2))--(2,0)--cycle,cyan);
draw(graph(f,0,2.5),blue+1.2, "$y=5x$");
 
xaxis("$x$",BottomTop,LeftTicks);
yaxis("$y$",LeftRight,RightTicks(trailingzero));
 
label("$y=5x$",(1.2,8),black);
 
pen p = white;
shipout(bbox(p,Fill));

$$\begin{array}{rlr}& {\int }_2^1\pi r^{2}dx& \\ & & \\ & {\int }_2^1\pi (5x{)}^{2}dx& \\ & & \\ & {\int }_2^1\pi 25x^{2}dx& \\ & & \\ & {\int }_2^1\pi 25x^{2}dx& \\ & & \\ & =\frac{25\pi }{3}{x}^3{|}_2^1& \\ & & \\ & =\frac{25\pi }{3}-\frac{25\pi }{3}{2}^3& \\ & & \\ & =\frac{25\pi }{3}-\frac{8(25\pi )}{3}& \\ & & \\ & =\frac{25\pi }{3}-\frac{200\pi }{3}& \\ & & \\ & =-\frac{175\pi }{3}& \\ & & \\ & =\frac{175\pi }{3}& \text{absolute value as if I got the integrals in the right order}\end{array}$$

7.3 6

$$\begin{array}{rl}& {\int }_1^2\pi h2rdr\\ & \\ & {\int }_1^2\pi h2xdx\\ & \\ & {\int }_1^2\pi 5x2xdx\\ & \\ & {\int }_1^2\pi 10x^{2}dx\\ & \\ & =\frac{10\pi }{3}{x}^3{|}_1^2\\ & \\ & =\frac{10\pi }{3}{2}^3-\frac{10\pi }{3}\\ & \\ & =\frac{80\pi }{3}-\frac{10\pi }{3}\\ & \\ & =\frac{70\pi }{3}\end{array}$$