A landscape architect is designing a decorative concrete partition for a park. The height of the partition, HHH metres, at a horizontal distance ddd metres from a central pillar, is modeled by the equation
H=24d+9,d≥−2.25 H = 2\sqrt{4d + 9}, \quad d \ge -2.25 H=24d+9,d≥−2.25A straight support beam, represented by line lll, is attached to the partition at the point P(4,10)P(4, 10)P(4,10). The beam is perpendicular to the curve of the partition at PPP.
Use calculus to show that an equation for the line lll is
5d+4H−60=0 5d + 4H - 60 = 0 5d+4H−60=0The region RRR is the cross-sectional area of the partition's side, bounded by the curve, the ground (the line H=0H = 0H=0), and the support beam lll.
Use algebraic integration to find the exact area of RRR.
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.