Calculus Examples

Evaluate the Integral integral from -3 to 0 of 1+ square root of 9-x^2 with respect to x
Step 1
Split the single integral into multiple integrals.
Step 2
Apply the constant rule.
Step 3
Let , where . Then . Note that since , is positive.
Step 4
Simplify terms.
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Step 4.1
Simplify .
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Step 4.1.1
Simplify each term.
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Step 4.1.1.1
Apply the product rule to .
Step 4.1.1.2
Raise to the power of .
Step 4.1.1.3
Multiply by .
Step 4.1.2
Factor out of .
Step 4.1.3
Factor out of .
Step 4.1.4
Factor out of .
Step 4.1.5
Apply pythagorean identity.
Step 4.1.6
Rewrite as .
Step 4.1.7
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2
Simplify.
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Step 4.2.1
Multiply by .
Step 4.2.2
Raise to the power of .
Step 4.2.3
Raise to the power of .
Step 4.2.4
Use the power rule to combine exponents.
Step 4.2.5
Add and .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Use the half-angle formula to rewrite as .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Combine and .
Step 9
Split the single integral into multiple integrals.
Step 10
Apply the constant rule.
Step 11
Let . Then , so . Rewrite using and .
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Step 11.1
Let . Find .
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Step 11.1.1
Differentiate .
Step 11.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 11.1.3
Differentiate using the Power Rule which states that is where .
Step 11.1.4
Multiply by .
Step 11.2
Substitute the lower limit in for in .
Step 11.3
Cancel the common factor of .
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Step 11.3.1
Move the leading negative in into the numerator.
Step 11.3.2
Cancel the common factor.
Step 11.3.3
Rewrite the expression.
Step 11.4
Substitute the upper limit in for in .
Step 11.5
Multiply by .
Step 11.6
The values found for and will be used to evaluate the definite integral.
Step 11.7
Rewrite the problem using , , and the new limits of integration.
Step 12
Combine and .
Step 13
Since is constant with respect to , move out of the integral.
Step 14
The integral of with respect to is .
Step 15
Substitute and simplify.
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Step 15.1
Evaluate at and at .
Step 15.2
Evaluate at and at .
Step 15.3
Evaluate at and at .
Step 15.4
Simplify.
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Step 15.4.1
Add and .
Step 15.4.2
Add and .
Step 16
Simplify.
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Step 16.1
The exact value of is .
Step 16.2
Subtract from .
Step 16.3
Combine and .
Step 17
Simplify.
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Step 17.1
Simplify the numerator.
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Step 17.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 17.1.2
The exact value of is .
Step 17.2
Divide by .
Step 17.3
Multiply by .
Step 17.4
Add and .
Step 17.5
Multiply .
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Step 17.5.1
Multiply by .
Step 17.5.2
Multiply by .
Step 18
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 19