3.7 Q-3

Question Statement

Find the area below the curve $y = 3 \sqrt{x}$ and above the $x$-axis, between $x = 1$ and $x = 4$.

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Background and Explanation

This problem requires the use of definite integration to find the area under the curve $y = 3 \sqrt{x}$ between the limits $x = 1$ and $x = 4$.

We need to evaluate the integral of the function $3 \sqrt{x}$ within the given limits. The curve is in the first quadrant, and it intersects the $x$-axis at $x = 0$.

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Solution

We want to compute the following integral to find the area:

$$A = \int_{1}^{4} 3 \sqrt{x} , dx$$

Step 1: Express the integral in a simpler form

We can rewrite the integrand as $3 \times x^{1/2}$:

$$A = 3 \int_{1}^{4} x^{1/2} , dx$$

Step 2: Apply the power rule of integration

The integral of $x^{n}$ is given by:

$$\int x^n , dx = \frac{x^{n+1}}{n+1}$$

Using this rule, we can now integrate $x^{1/2}$:

$$\int x^{1/2} , dx = \frac{x^{3/2}}{3/2} = \frac{2x^{3/2}}{3}$$

Step 3: Evaluate the integral

Now, we evaluate the integral from $x = 1$ to $x = 4$:

$$A = 3 \left[ \frac{2x^{3/2}}{3} \right]_{1}^{4}$$

This simplifies to:

$$A = 2 \left[ x^{3/2} \right]_{1}^{4}$$

Now, evaluate $x^{3/2}$ at the limits $x = 4$ and $x = 1$:

$$A = 2 \left[ (4)^{3/2} - (1)^{3/2} \right]$$

Since $4^{3/2} = (2^2)^{3/2} = 2^3 = 8$ and $1^{3/2} = 1$, we get:

$$A = 2 \left[ 8 - 1 \right] = 2 \times 7 = 14$$

Thus, the area is 14 square units.

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Key Formulas or Methods Used

• Definite Integral: The area under the curve from $x = a$ to $x = b$ is given by:

$$A = \int_{a}^{b} f(x) , dx$$

• Power Rule for Integration: The integral of $x^n$ is:

$$\int x^n , dx = \frac{x^{n+1}}{n+1}$$

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Summary of Steps

1. Set up the integral: $A = 3 \int_{1}^{4} x^{1/2} , dx$

2. Use the power rule to integrate: $\int x^{1/2} , dx = \frac{2x^{3/2}}{3}$

3. Evaluate the definite integral: $A = 2 \left[ x^{3/2} \right]_{1}^{4}$

4. Simplify the expression: $A = 2 \left[ 8 - 1 \right] = 14$

5. The area is 14 square units.