How to find total area of a region

Question

We have a parabola whose formula is y=5sin^2x and is bounded at the top by the line y=5. It is shaded under the y=5 line between x=pi/2 and x=3pi/2. How do I find the area of this shaded region?

 

I have a similar problem where y=4(sinx)√1+cosx. I'm supposed to find its area between 0 and pi. 

 

Thank you! 

Answer 1

Strategy:

 

Look at the graph of f(x)=5sin²x and notice the symmetry around x=π. If you can calculate the area between f(x) and the x-axis on the interval [π/2,π] you're set. Just subtract twice this area from the area of the rectangle formed between the x-axis and y=5 on the interval x ε [π/2, 3π/2].

 

All you need now is an antiderivative for f(x) and some skills with the algebra of definite integrals.

 

Good luck!

 

p.s. I think you might have a typo in your second question. Perhaps some additional parenthesis needed?