A landscape architect is designing a feature for a park. The cross-sectional elevation of the terrain is modelled by a curve CCC with equation
y=x12(15−x),x≥0 y = x^{\frac{1}{2}}(15 - x), \quad x \ge 0 y=x21(15−x),x≥0where yyy is the height in decametres and xxx is the horizontal distance in decametres. The curve CCC has a stationary point at PPP.
Using calculus, determine the xxx-coordinate of PPP.
The region R1R_1R1, representing a hill section, is bounded by the curve CCC and the xxx-axis.
The region R2R_2R2, representing a valley section, is bounded by the curve CCC, the xxx-axis, and the vertical line x=kx = kx=k, where k>15k > 15k>15 is a constant.
Given that the area of the hill R1R_1R1 is equal to the area of the valley R2R_2R2, find, using calculus, the exact value of kkk.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.