Solve ∫ (from 0 to 3) of sqrt{4-x^2 wrt y

$$\int_{ 0 }^{ 3 } \sqrt{ 4- { x }^{ 2 } } d y$$

Evaluate

$$3\sqrt{4-x^{2}}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{4-x^{2}}\mathrm{d}y$$

Find the integral of \sqrt{4-x^{2}} using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\sqrt{4-x^{2}}y$$

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.

$$\left(4-x^{2}\right)^{\frac{1}{2}}\times 3-\left(4-x^{2}\right)^{\frac{1}{2}}\times 0$$

Simplify.

$$3\sqrt{4-x^{2}}$$

Differentiate w.r.t. x

$$-\frac{3x}{\sqrt{4-x^{2}}}$$