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Area Under the Curve
Vocabulary
Fill in the following chart:
Word  Definition 
_________________  Used to calculate the area under a curve. It has many applications in science, including finding distance traveled by an object moving at inconstant speeds. 
Fundamental Theorem of Calculus  __________________________________________________________________ 
_________________  the function whose derivative is the function you want the antiderivative of 
Riemann Sums
How do you find the area of a rectangle? ________________________
To find the area under a curve you can fill the curve with many rectangles. As the width of the rectangles decrease, the area becomes more and more exact. In other words, the area under the curve is the limit of the total area of the rectangles as the widths of the rectangles approach zero.
Using rectangles to find the area under a curve is called a Riemann Sum. Riemann Sums can be left, right, or middle depending on which part of the rectangle touches the curve. The figure above uses a right Riemann Sum.
The figure below uses a left Riemann Sum.
Note: If the rectangles are below the yaxis it creates a negative area.
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Find the area between the curve and the xaxis:
 Curve y = x on the interval x = 1 to x = 3.
 Curve y =  x from x = 1 to x = 3.
 Curve y = x from x = 3 to x = 3
 Approximate the area under \begin{align*}y = 2x  4\end{align*} on the interval [0,3] using the middle Riemann Sum for y with 6 subintervals.
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Definite Integrals.
Finding the area under a curve by making the width of the rectangles approach zero is called the definite integral.
The Definite Integral (The Limit Method)


To find the definite integral, you take the antiderivative, or the opposite of the derivative, of the function.
Definition: The Antiderivative


Try to logically think backwards to find the antiderivative, and always check to make sure the derivative of your antiderivative is the function!
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Here are some rules for antiderivation:
Rules of Finding the Antiderivatives of Power Functions



The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus


Using the Fundamental Theorem, evaluate the folloiwng:
 \begin{align*}\int_{1}^{2} x^2 dx =\end{align*}
 \begin{align*}\int (4x^4  x^2 + 3)dx =\end{align*}
 \begin{align*}\int_{1}^{4} \sqrt{x} dx =\end{align*}
 \begin{align*}\int_{5}^{7} \frac{dx}{x} =\end{align*}
 \begin{align*}\int_{p}^{\frac{3p}{2}}6sin(x) dx =\end{align*}