1. Ch. 6 – The Definite Integral
6.2 – Definite Integrals

2. Ex: Set up various RAMs to estimate the area of the region bounded by the graph of f(x) = 4x3 – 2x over [0, 1] .
Note: Area under the x-axis has a negative value because the y-values are negative!
Our goal is to get the most accurate estimate
Let’s try using LRAM with 10 subintervals…

3. Ex: Set up various RAMs to estimate the area of the region bounded by the graph of f(x) = 4x3 – 2x over [0, 1] .
Note: Area under the x-axis has a negative value because the y-values are negative!
Our goal is to get the most accurate estimate
Now we’ll try LRAM with 20 subintervals…

4. Ex: Set up various RAMs to estimate the area of the region bounded by the graph of f(x) = 4x3 – 2x over [0, 1] .
Now we’ll use an infinite number of subintervals. This is the best estimate!
Since LRAM, RRAM, and MRAM approach the same value, it doesn’t matter which one we use.
Since each width is so tiny, we’ll say each width is dx
Let each y-value be written as f(xk) to represent the kth subinterval on x

5. The Definite Integral
This represents the integral of f from a to b
The curvy S stands for an infinite sum
a = lower bound of integration
b = upper bound of integration
f(x) = the integrand (function to be integrated)
dx = tells you with respect to which variable you integrate
Evaluate this integral to find the area under the curve y = f(x) over [a, b]!

6. Ex: Write the following limits as definite integrals.

7. Ex: Evaluate the following integrals using the figure below
Ex: Evaluate the following integrals using the figure below. Assume all axes are scaled by ones.
Graph of f

8. Ex: Evaluate the following integrals using the figure below
Ex: Evaluate the following integrals using the figure below. Assume all axes are scaled by ones.
Graph of f

9. Ex: Use the graph of the integrand to evaluate the integral.
Find the area under the curve!
Shape under curve is a semicircle
Area = (area above x-axis) – (area below x-axis)
Each shape is a triangle
Find the x-intercept first to find the base lengths

10. Ex: Evaluate algebraically:
To evaluate a definite integral, first find the antiderivative, then evaluate it at the limits of integration
Ex: Evaluate algebraically:
What’s the antiderivative of 2x? Remember, for polynomials, just add 1 to the exponent and divide by that new exponent…
Antiderivative =
Now evaluate from -1 to 4…
Note: All continuous functions are integrable (can be integrated).

11. Ex: Use your calculator to evaluate:
Go to the MATH menu on your calculator and scroll to the bottom to pull up the fnInt function
fnInt takes 4 arguments: the function, the variable you’re integrating, the lower limit, and the upper limit
The entry should look like this:
Answer ≈.873
Be careful with your parentheses!

12. Ex: Use the graph of the integrand to evaluate the integral.