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Area Under the Curve

Integrals and Riemann Sums.

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Area Under the Curve and the Fundamental Theorem of Calclulus

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Area Under the Curve


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 y-axis it creates a negative area.


Find the area between the curve and the x-axis:

  1. Curve on the interval x = 1 to x = 3.
  2. Curve = - from = 1 to = 3.
  3. Curve from = -3 to = 3
  4. 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)
The area between a curve ) and the -axis over the interval [ ] can be calculated by
\begin{align*}A = \int_{a}^{b} f(x) dx = \lim_{n \rightarrow \infty} \sum_{i = 1}^n f(x_i) \Delta x\end{align*}
\begin{align*}\Delta x = \frac{b - a} {n}\end{align*}
is the width of the subintervals.

To find the definite integral, you take the antiderivative, or the opposite of the derivative, of the function.

Definition: The Antiderivative
If ' ( ) = ), then '( ) is said to be the antiderivative of ).

Try to logically think backwards to find the antiderivative, and always check to make sure the derivative of your antiderivative is the function!


Here are some rules for antiderivation:

Rules of Finding the Antiderivatives of Power Functions
  • The Power Rule
\begin{align*}\int x^n dx = \frac{1} {n + 1} x^{n + 1} + C\end{align*}
where is constant of integration and is a rational number not equal to -1.
  • A Constant Multiple of a Function Rule
\begin{align*}\int k x^n dx = k \int x^n dx = k \cdot \frac{1} {n + 1} x^{n + 1} + C\end{align*}
where is a constant.
  • Sum and Difference Rule
\begin{align*}\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx\end{align*}
  • The Constant rule
\begin{align*}\int k \cdot dx = kx + C\end{align*}
where is a constant. (Notice that this rule comes as a result of the power rule above.)

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus
If a function ) is defined over the interval [ ] and if ) is the antidervative of on [ a, ], then
\begin{align*}\int_{a}^{b} f(x) dx = F(x)|^b_a\end{align*}
\begin{align*}= F(b) - F(a)\end{align*}
Using the Fundamental Theorem, evaluate the folloiwng:
  1.  \begin{align*}\int_{1}^{2} x^2 dx =\end{align*}
  2. \begin{align*}\int (4x^4 - x^2 + 3)dx =\end{align*}
  3.  \begin{align*}\int_{1}^{4} \sqrt{x} dx =\end{align*}
  4. \begin{align*}\int_{5}^{7} \frac{dx}{x} =\end{align*}
  5. \begin{align*}\int_{-p}^{\frac{3p}{2}}6sin(x) dx =\end{align*}
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