Question

Question #e31c1

Answer 1

Please see the explanation.

Explanation:

Here is a graph of the 3 boundaries:

Decide whether to integrate `dy` or `dx`:

You want to find the area between the y axis and green curve until it intersects with the blue curve. At first, you might want to do it with respect to y but you will be integrating inverse sine and inverse cosine function. I say no. I think that it is easier to find the area of the same region as the difference between the functions with respect to x.

Find the limits of integration:

We know that we want to start integrating from `x = 0` but we need to find where we stop by finding the x coordinate of the point of intersection. Set the right side of the two equations equal to each other:

`6cos(x) = 7sin(2x)`

Use `sin(2x) = 2sin(x)cos(x)`

`6cos(x) = 14sin(x)cos(x)`

`6 = 14sin(x)`

`sin(x) = 3/7`

`x = sin^-1(3/7)`

`x ~~ 0.4429`

Here is your integral is:

`Area = int_0^0.4429(6cos(x) - 7sin(2x))dx`

`Area = 6sin(x) + 7/2cos(2x)|_0^0.4429`

`Area ~~ 1.28571`

I hope that this helps.