The profile of a cooling fin in a high-performance engine is modeled by a curve CCC in the (x,y)(x, y)(x,y) plane. For y>0y > 0y>0, the gradient of the curve at any point (x,y)(x, y)(x,y) is given by the differential equation
dydx=x3y227 \frac{dy}{dx} = \frac{x^3 y^2}{27} dxdy=27x3y2The curve CCC passes through the point with coordinates (3,1)(3, 1)(3,1). Show that CCC intersects the coordinate axes at exactly one point and state the coordinates of this point. Fully justify your answer.
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.