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\int _{3}^{9}-6y-54\mathrm{d}y
Calculate \sqrt{-6y-54} to the power of 2 and get -6y-54.
\int -6y-54\mathrm{d}y
Evaluate the indefinite integral first.
\int -6y\mathrm{d}y+\int -54\mathrm{d}y
Integrate the sum term by term.
-6\int y\mathrm{d}y+\int -54\mathrm{d}y
Factor out the constant in each of the terms.
-3y^{2}+\int -54\mathrm{d}y
Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}. Multiply -6 times \frac{y^{2}}{2}.
-3y^{2}-54y
Find the integral of -54 using the table of common integrals rule \int a\mathrm{d}y=ay.
-3\times 9^{2}-54\times 9-\left(-3\times 3^{2}-54\times 3\right)
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.
-540
Simplify.