As always, whenever we find the area between two curves, we need to perform each of the following steps.
1. Sketch the bounded region.
2. Draw a representative rectangle.
3. Since the width of the rectangle will be parallel to the x-axis for now, label the width as dx.
4. Label the height of the rectangle as the top function minus the bottom function. In other words, if f (x) is
the top funciton and if g (x) is the bottom function, we would label the height as f (x) - g(x).
5. Identify the left bound x = a and the right bound x = b for the area. (Either bound could be the x-value
where the two curves intersect, or either could be a vertical line.)
6. Set up and use the integral formula Area = ∫[f(x) - g(x)] dx to find the area of the bounded region.
Question 1
Find the area of the region bounded by the graphs of
â– y = x + 1,
y = 9-x²,
x = -1, and
x = 2.
Question 2
Find the area of the region bounded by f (x) = cos x and g (x) = sin x on the interval [Ï€/4, 5Ï€/4].
Question 3
Find the area of the region bounded by
y = √x,
y = 2x², and
• y = -√2x.