Mathematics

Problem

The region

R

is bounded by the parabolas

y = x^{2}

and

y = 6x - x^{2}

together with the

y

-axis

(x = 0)

and the vertical line

x = 5

. a. (i) Verify algebraically that the two parabolas intersect at

x = 0

and

x = 3

only. b. (ii) Draw a rough sketch of the two parabolas, the

y

-axis, and the line

x = 5

. Shade the region

R

. c. (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

. d. (iv) Evaluate the integral(s) from part (iii) to obtain the exact area of

R

. Leave your answer in simplest form.

Intuitive explanation

The key is to first find where these two parabolas intersect, then identify which curve is on top in each interval. Once you sketch the region, you'll see it may split into separate pieces—some bounded by one parabola pair, others by a parabola and a vertical line. Set up the area integral by integrating the difference of the upper and lower functions with respect to $x$ , being careful to use the correct bounds and function order for each section.

Intuitive explanation

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