3.7 Q-1

Question Statement

Find the area between the $x$ axis and the curve $y = x^2 + 1$ from $x = 1$ to $x = 2$.

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Background and Explanation

To solve this problem, we need to calculate the area under the curve $y = x^2 + 1$ between $x = 1$ and $x = 2$. The concept used here is definite integration, which gives the area under a curve between two points.

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Solution

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

$$A = \int_{1}^{2} (x^2 + 1) , dx$$

This integral can be broken into two simpler integrals:

$$A = \int_{1}^{2} x^2 , dx + \int_{1}^{2} 1 , dx$$

Step 1: Evaluate the first integral

The integral of $x^2$ is:

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

Now evaluate this from $x = 1$ to $x = 2$:

$$\left[ \frac{x^3}{3} \right]_{1}^{2} = \frac{(2)^3}{3} - \frac{(1)^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$$

Step 2: Evaluate the second integral

The integral of 1 is simply:

$$\int 1 , dx = x$$

Now evaluate this from $x = 1$ to $x = 2$:

$$\left[ x \right]_{1}^{2} = 2 - 1 = 1$$

Step 3: Combine the results

Now, add the results from the two integrals:

$$A = \frac{7}{3} + 1 = \frac{7}{3} + \frac{3}{3} = \frac{10}{3} \text{ square units}$$

Thus, the area under the curve is $\frac{10}{3}$ square units.

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Key Formulas or Methods Used

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

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

• Basic Integration Rules:
  • $\int x^n , dx = \frac{x^{n+1}}{n+1}$
  • $\int 1 , dx = x$

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Summary of Steps

1. Set up the integral: $A = \int_{1}^{2} (x^2 + 1) , dx$
2. Break the integral into two parts: $A = \int_{1}^{2} x^2 , dx + \int_{1}^{2} 1 , dx$
3. Evaluate the first integral: $\frac{7}{3}$
4. Evaluate the second integral: $1$
5. Add the results: $A = \frac{10}{3} \text{ square units}$