3.6 Q-4

Question Statement

Evaluate the integral: $\int_{-6}^{2} \sqrt{3 - x} , dx$

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Background and Explanation

In this problem, we are tasked with integrating a square root function. To solve this, we will first use substitution and then apply the power rule for integration. The square root function $\sqrt{3 - x}$ can be expressed as $(3 - x)^{1/2}$, which is easier to integrate using standard formulas.

We need to recognize that a substitution may simplify the expression for easier integration.

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Solution

1. Rewrite the Integral:

   The given integral is: $\int_{-6}^{2} \sqrt{3 - x} , dx$

   First, notice that the function inside the square root is $(3 - x)$, so we can use a substitution to simplify the integral.

2. Apply Substitution:

   Let: $u = 3 - x$
   Then: $du = -dx$

   Now substitute into the integral. The limits change accordingly:

   • When $x = -6$, $u = 3 - (-6) = 9$.
   • When $x = 2$, $u = 3 - 2 = 1$.

   The integral becomes: $\int_{-6}^{2} (3 - x)^{1/2} , dx = - \int_{9}^{1} u^{1/2} , du$

   The negative sign comes from $du = -dx$, which flips the limits of integration.

3. Integrate:

   Now apply the power rule for integration. For $u^{1/2}$, we have: $\int u^{1/2} , du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}$

   Therefore, the integral becomes: $- \left[ \frac{2}{3} u^{3/2} \right]_{9}^{1}$

4. Evaluate the Integral:

   Now evaluate the expression at the limits $u = 1$ and $u = 9$: $- \left[ \frac{2}{3} (1)^{3/2} - \frac{2}{3} (9)^{3/2} \right]$

   Simplify: $= - \frac{2}{3} \left[ 1 - (9)^{3/2} \right]$

   Since $9^{3/2} = 27$, we get: $= - \frac{2}{3} \left[ 1 - 27 \right] = - \frac{2}{3} \times (-26)$

5. Final Calculation:

   Simplify further: $= \frac{52}{3}$

Thus, the value of the integral is: $\boxed{\frac{52}{3}}$

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Key Formulas or Methods Used

• Substitution:
  $u = 3 - x$, leading to $du = -dx$.

• Power rule for integration:
  $\int u^n , du = \frac{u^{n+1}}{n+1}$

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Summary of Steps

1. Set up the substitution:
   $u = 3 - x$, $du = -dx$.

2. Change the limits of integration accordingly:
   $- \int_{9}^{1} u^{1/2} , du$

3. Apply the power rule:
   $- \frac{2}{3} u^{3/2}$

4. Evaluate the integral at the limits $u = 1$ and $u = 9$: $- \frac{2}{3} \left[ 1 - 27 \right]$

5. Simplify the final result: $\frac{52}{3}$

Thus, the final result is $\frac{52}{3}$.