Evaluate the Integral

Problem

(∫_0^7)(√(,y+9)*d(y))

Solution

1. Identify the integral as a definite integral of a radical function.

2. Rewrite the square root as a fractional exponent to make it easier to apply the power rule.

(∫_0^7)((y+9)(1/2)*d(y))

3. Apply the power rule for integration, which states that (∫_^)(un*d(u))=(u(n+1))/(n+1) Since the derivative of y+9 is 1 no complex substitution is required.

[((y+9)(3/2))/(3/2)]70

4. Simplify the constant coefficient by multiplying by the reciprocal.

[2/3*(y+9)(3/2)]70

5. Substitute the upper limit y=7 and the lower limit y=0 into the expression.

2/3*(7+9)(3/2)−2/3*(0+9)(3/2)

6. Evaluate the numerical values by calculating the square roots first.

2/3*(16)(3/2)−2/3*(9)(3/2)

7. Calculate the powers where 16(3/2)=(4)3=64 and 9(3/2)=(3)3=27

2/3*(64)−2/3*(27)

8. Subtract the resulting values to find the final area.

128/3−54/3=74/3

Final Answer

(∫_0^7)(√(,y+9)*d(y))=74/3

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