Compute the following sum: $$\sum \limits _{k=\sage {p1indexStart}}^{\sage {p1indexEnd}} \sage {p1f1(x=k)} = \answer {\sage {p1ans}}$$

Compute the indefinite integral of $f(x) = \sage {p3f1}$: $$\int \sage {p3f1}dx = \answer [validator=sameDerivative]{\sage {p3ans}}$$

Use limits to find a perfect approximation for the area under the curve of the function $f(x) = \sage {p4f1}$ from $x=\sage {p4start}$ to $x=\sage {p4end}$. $\answer {\sage {p4ans}}$.

Compute the indefinite integral of $f(x) = \sage {p5fdisp}$: $$\int \sage {p5fdisp}dx = \answer [validator=sameDerivative]{\sage {p5f1}}$$

• The $\int \limits _{\sage {p9c1}}^{\sage {p9c2}}$ means:
  We want the indefinite integral of $f(x)$. We want the derivative of $f(x)$. We want the unsigned area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c1}$ to $x=\sage {p9c2}$. We want the unsigned area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c2}$ to $x=\sage {p9c1}$. We want the signed area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c1}$ to $x=\sage {p9c2}$. We want the signed area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c2}$ to $x=\sage {p9c1}$. We want to integrate with respect to the $x$ variable. We want to integrate with respect to the $d$ variable. The integrand, which is the function whose graph we use in the integration process.

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• The $dx$ means:
  We want the indefinite integral of $f(x)$. We want the derivative of $f(x)$. We want the unsigned area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c1}$ to $x=\sage {p9c2}$. We want the unsigned area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c2}$ to $x=\sage {p9c1}$. We want the signed area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c1}$ to $x=\sage {p9c2}$. We want the signed area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c2}$ to $x=\sage {p9c1}$. We want to integrate with respect to the $x$ variable. We want to integrate with respect to the $d$ variable. The integrand, which is the function whose graph we use in the integration process.

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• The $f(x)$ is:
  We want the indefinite integral of $f(x)$. We want the derivative of $f(x)$. We want the unsigned area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c1}$ to $x=\sage {p9c2}$. We want the unsigned area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c2}$ to $x=\sage {p9c1}$. We want the signed area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c1}$ to $x=\sage {p9c2}$. We want the signed area between the $x$ axis and the graph of $f(x)$ from $x=\sage {p9c2}$ to $x=\sage {p9c1}$. We want to integrate with respect to the $x$ variable. We want to integrate with respect to the $d$ variable. The integrand, which is the function whose graph we use in the integration process.

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Use $\sage {p10RecNum}$ rectangles to approximate the area under the curve of the function $f(x) = \sage {p10f1}$ from $x=\sage {p10start}$ to $x=\sage {p10end}$ using...

• Right Endpoint Approximation Method: $\answer {\sage {p10ansREP}}$
• Left Endpoint Approximation Method: $\answer {\sage {p10ansLEP}}$
• Midpoint Approximation Method: $\answer {\sage {p10ansMP}}$