The region is bounded by the parabolas and together with the -axis and the vertical line .
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
The region R is bounded by the parabolas y=x2 and y=6x−x2 together with the y-axis (x=0) and the vertical line x=5.
(i) Verify algebraically that the two parabolas intersect at x=0 and x=3 only.
(ii) Draw a rough sketch of the two parabolas, the y-axis, and the line x=5. Shade the region R.
(iii) Split R into suitable sub-regions so that in each sub-region a single curve acts as the upper boundary. Write the corresponding definite integral(s) that give the area of R.
(iv) Evaluate the integral(s) from part (iii) to obtain the exact area of R. Leave your answer in simplest form.