An artist is designing a stained-glass panel. The boundary of the panel is modeled by the curve $C$ with equation $y = 6x - x^2$ , for $\ge 0$ . A decorative lead strip is placed along the line $l$ with equation $y = k x$ , where $k$ is a constant and $0 < k < 6$ .

The line $l$ and the curve $C$ intersect at the origin $O$ and at the point $P$ .

Find, in terms of $k$ , the coordinates of $P$ .

The region $R_1$ is bounded by the curve $C$ and the line $l$ . Show that the area of $R_1$ is

\frac{(6 - k)^3}{6}

The region $R_2$ is bounded by the curve $C$ , the $x$ -axis, and the line $l$ , such that $R_1$ and $R_2$ together comprise the total area under the curve for $\le x \le 6$ . Given that the area of $R_1$ is equal to the area of $R_2$ , find the exact value of $k$ .

The line $l$ and the curve $C$ intersect at the origin $O$ and at the point $P$ .

Find, in terms of $k$ , the coordinates of $P$ .

The region $R_1$ is bounded by the curve $C$ and the line $l$ . Show that the area of $R_1$ is

\frac{(6 - k)^3}{6}

Integration

Question 509

Markscheme

Integration

Question 509

Markscheme

Integration

Question 509

Integration Questions

Integration

Question 509

Integration Questions