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find the area under the graph of each function over the given

y=x^(2)+x+1quad[2,3]

find the area under the graph of each function over the given $\newline$ $y=x^{2}+x+1 \quad[2,3]$

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Find derivatives using the chain rule I

Full solution

Q. find the area under the graph of each function over the given

\newline

y=x^{2}+x+1 \quad[2,3]

Set up integral: To find the area, we need to integrate the function from $2$ to $3$ . So, let's set up the integral: $\int_{2}^{3} (x^2 + x + 1) \, dx$ .
Integrate $x^2$ : First, integrate $x^2$ . The antiderivative of $x^2$ is $(1/3)x^3$ .
Integrate $x$ : Next, integrate $x$ . The antiderivative of $x$ is $\frac{1}{2}x^2$ .
Integrate $1$ : Finally, integrate $1$ . The antiderivative of $1$ is $x$ .
Add antiderivatives: Now, add all the antiderivatives together to get the antiderivative of the function: $\frac{1}{3}x^3 + \frac{1}{2}x^2 + x$ .
Evaluate upper limit: Evaluate the antiderivative from $2$ to $3$ . Plug in the upper limit: $(\frac{1}{3})(3)^3 + (\frac{1}{2})(3)^2 + (3)$ .
Calculate $x=3$ : Calculate the value for $x = 3$ : $\left(\frac{1}{3}\right)(27) + \left(\frac{1}{2}\right)(9) + 3 = 9 + 4.5 + 3$ .
Evaluate lower limit: Now plug in the lower limit: $(\frac{1}{3})(2)^3 + (\frac{1}{2})(2)^2 + (2)$ .
Calculate $x=2$ : Calculate the value for $x = 2$ : $\left(\frac{1}{3}\right)(8) + \left(\frac{1}{2}\right)(4) + 2 = 2.666... + 2 + 2$ .
Subtract values: Subtract the lower limit value from the upper limit value: $(9 + 4.5 + 3) - (2.666\ldots + 2 + 2)$ .
Find the area: Do the subtraction to find the area: $16.5 - 6.666\ldots = 9.833\ldots$ .

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