Calculating area -Calculus (disk/washer)

Question

Consider the region R in the first quadrant below the curve y=cos(x), above the curve y=sin(x) and to the right of the y axis.

a) Find the x and y coordinates of the intersection point of the 2 curves.

b) calculate the area of region R.

C) Using the disks/ washer method write the integral that calculates the volume of the solid obtained by rotating the region R around the x axis. Don’t evaluate

D) C) Using either the disks/ washer method write the integral that calculates the volume of the solid obtained by rotating the region R around the y axis. Don’t evaluate

Answer 1

We use I[a, b] for the integral from a to b and E[a, b] for evaluation from a to b

a) sin and cos are equal, (1/√2), at 45⁰= π/4

b) I[0, π/4] cos x - sin x dx = sin x + cos x E[0, π/4] = sin π/4 +cos π/4 - sin 0 - cos 0 = 1/√2 + 1/√2 - 0 - 1 =

√2 - 1

c) We use the washer method for rotation about the x-axis

I[0, π/4] π cos²x - π sin²x dx =

I[0, π/4] π cos 2x dx = π/2 sin 2x E[0, π/4] = π/2 sin π/2 - π/2 sin 0 = π/2 * 1 - π/2* 0 = π/2

d) We use the disk method for rotation about the y-axis

I[0, π/4] 2 π x (cos x - sin x) dx = 2 π (x sin x + cos x + x cos x - sin x) E[0, π/4]] =

2 π(π/4 sin π/4 + cos π/4 + π/4cos π/4 - sin π/4 - (0 sin 0 +cos 0 + 0 cos 0 - sin 0)) =

2 π(π/4(1/√2 + 1/√2) - 1/√2 + 1/√2 - 1) =

π² √2/2 - 2 π

Answer 2

Below is the link to an image of the work

https://ibb.co/9gd2vTb

Let me know if you need any clarification or explanation.