Find the Area Under the Curve y=x^2+2 , [0,1]

Solution:

Step 1: Determine the intersection points of the curves

Since we are given the range [0,1], we do not need to find intersections for this problem.

Step 2: Calculate the area under the curve

To find the area under the curve $y = x^2 + 2$ from $x = 0$ to $x = 1$, we need to set up the integral:

$$\text{Area} = \int_{0}^{1} (x^2 + 2) \, dx$$

Step 3: Perform the integration

We will integrate the function term by term:

$$\int_{0}^{1} x^2 \, dx + \int_{0}^{1} 2 \, dx$$

Using the power rule for integration:

$$\left[\frac{x^3}{3}\right]_0^1 + \left[2x\right]_0^1$$

Step 4: Evaluate the integrals

Calculate the definite integrals by substituting the upper and lower limits:

$$\left(\frac{1^3}{3} + 2 \cdot 1\right) - \left(\frac{0^3}{3} + 2 \cdot 0\right)$$

Simplify the expression:

$$\frac{1}{3} + 2 - 0$$

Step 5: Find the area

The area under the curve is:

$$\text{Area} = \frac{1}{3} + 2 = \frac{7}{3}$$

Knowledge Notes:

To solve the problem of finding the area under a curve, we use definite integration. Here are the relevant knowledge points:

1. Definite Integral: The definite integral of a function between two limits is the net area under the curve of the function between those limits. It is denoted as $\int_{a}^{b} f(x) \, dx$, where $a$ and $b$ are the lower and upper limits, respectively.

2. Power Rule for Integration: When integrating a power of $x$, the formula is $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n$ is a real number and $C$ is the constant of integration. For definite integrals, we evaluate the result at the upper and lower limits and subtract.

3. Integration of a Constant: The integral of a constant $a$ with respect to $x$ over an interval $[b,c]$ is $a(c - b)$.

4. Evaluating Definite Integrals: After finding the antiderivative, we substitute the upper limit into the antiderivative, subtract the result of substituting the lower limit, and simplify to find the definite integral's value.

5. Complex Numbers: In the original solution, the equation $x^2 + 2 = 0$ was mistakenly solved, leading to complex solutions. However, for the area under the curve problem, we only consider real values of $x$ in the interval [0,1], so this step was unnecessary and incorrect for the context of the problem.

6. Area Under the Curve: The area under the curve $y = f(x)$ from $x = a$ to $x = b$ is given by the definite integral $\int_{a}^{b} f(x) \, dx$. If $f(x)$ is above the $x$-axis in this interval, the area is positive; if $f(x)$ is below the $x$-axis, the area is negative.