Thread: area and volume bounded by y axis

1. area and volume bounded by y axis

Let area A be the region bounded by y=(2^x)-1, y=-2x+3, and the y axis.

a) find the exact value for the area of region A.

b) set up, but DONT integrate, an integral expression in terms of a signle variable for the volume of the solid generated when A is revolved around the x axis.

c) set up an integral expression in terms of a single variable for the volume of the solid generated when A is revolved around the y axis. Find an approximation for this volume correct to the nearest hundredth.

Ok, my first question is, since it is bounded by the y axis, i need to change the two equations to x=___________ ? I did that and for the y=(2^x)-1 iend up getting (log y-1)/log 2= x and when i try to find the area using a calculuator, it gives me an error sign... i know i must be doing something wrong but i don't know what.

basically that same issue i have carries over to b and c...

is it correct to use log?

thanks

2. Originally Posted by frog09
Let area A be the region bounded by y=(2^x)-1, y=-2x+3, and the y axis.

a) find the exact value for the area of region A.

b) set up, but DONT integrate, an integral expression in terms of a signle variable for the volume of the solid generated when A is revolved around the x axis.

c) set up an integral expression in terms of a single variable for the volume of the solid generated when A is revolved around the y axis. Find an approximation for this volume correct to the nearest hundredth.
a) find the intersection point ...

$2^x - 1 = 3 - 2x$

$2^x = 2(2 - x)$ ... $x = 1$

$A = \int_0^1 (2^x - 1) - (3 - 2x) \, dx$

$A = \int_0^1 2^x + 2x - 4 \, dx$

$A = \left[\frac{2^x}{\ln{2}} + x^2 - 4x\right]_0^1
$

you can evaluate the definite integral

b) method of washers ...

$V = \pi \int_0^1 (2^x-1)^2 - (3-2x)^2 \, dx$

c) method of cylindrical shells ...

$V = 2\pi \int_0^1 x(2^x + 2x - 4) \, dx$

3. i was making that much more complicated.

i have a question just in general,

When do you know to switch the equations, for examples to in terms of y??