Solve ∫ (from 0 to 2) of 9/64x^2y^2 wrt x

$$\int _ { 0 } ^ { 2 } \frac { 9 } { 64 } x ^ { 2 } y ^ { 2 } d x$$

Evaluate

$$\frac{3y^{2}}{8}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{9x^{2}y^{2}}{64}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{9y^{2}}{64}\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\frac{9y^{2}}{64}\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{3y^{2}x^{3}}{64}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{3}{64}y^{2}\times 2^{3}-\frac{3}{64}y^{2}\times 0^{3}$$

Simplify.

$$\frac{3y^{2}}{8}$$

Differentiate w.r.t. y

$$\frac{3y}{4}$$