The profile of a specialized parabolic drainage channel is modeled by the curve with equation
h(x)=2x2+256x−100,x>0 h(x) = 2x^2 + \frac{256}{\sqrt{x}} - 100, \quad x > 0 h(x)=2x2+x256−100,x>0where hhh is the depth in millimeters and xxx is the horizontal distance from the left edge of the sensor assembly. The point PPP is the minimum point on the curve.
Use calculus to show that the xxx-coordinate of PPP is 4.
A horizontal laser beam, represented by the line lll, passes through point PPP and is parallel to the xxx-axis. A shaded region RRR is bounded by the curve, the laser beam lll, and the vertical line x=1x = 1x=1.
Use algebraic integration to find the exact cross-sectional area of the region RRR.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
