Calculus Answers

1. Evaluate "\\intop"x²(1 + 2x³)³dx.

2. Evaluate "\\intop"xe^7x dx.

3. Find the volume of the solid of revolution when the curve y = 1 + x2 is revolved around the x-axis on [−2, 2].

Find the area of the paraboloid x2 + y2 = z inside the cylinder x2 + y2 = 9.

Calculate the area under the curve y=x3 +4x+1 from x=-3 to x=3.

Two gardens. A fence of length 100 ft is to be used to enclose two

gardens. One garden is to have a circular shape, and the other to be

square.

Determine how the fence should be cut so that the sum of the areas

inside both gardens is as large as possible.

Let f(x) = 5 + 12x -x^3. Using the differentiation techniques learnt, find

a. The intervals on which f is increasing

Find an equation of the tangent line to the curve 9x³ – y³ = 1 at the point (0, -1).

Find an equation of the tangent line to the curve x⁴ + 2y⁴ = 33 at the point (1,2).

3. a) Define tangent and normal of a curve with figure. Also find the equation of tangent and normal of the ellipse (x ^ 2)/4 + (y ^ 2)/16 = 1 at the point (- 1, 3) .

b) Explain maximum and minimum value of a function with graphically. Evaluate maximum and minimum value of the function f(x) = x ^ 3 - 3x ^ 2 + 3x + 1

∫ ௫ିଵ ௫ యି௫మିଶ௫ 𝑑𝑥 

1.       Find the function whose tangent has slope 4x + 1 for each value of x and whose graph passes through the point (1, 2).