5.2 The Definite Integral

Example: Evaluate an Integral Using the Definition Part 1 Use the definition of the definite integral to evaluate ∫x^2dx with a = 0 and b = 2. Use a right-endpoint approximation to generate the Riemann sum.

1. Find ∆x 2. Find xi 3. Substitute the value from step two into x. 4. Use the value from step 3 and right it in sigma notation and use the appropriate summation formula 5. Calculate the definite integral by taking the limit as n approaches infinity

Example: Find the total area between the function f(x) = 2x and the x-axis over the interval [-3. 3]

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Applications of the Definite Integral

• finding displacement when given a velocity function v(t). • If v(t) represents the velocity of an object as a function of time, then the area under the curve tells us how far the object is from its original position. • When velocity is a constant, the area under the curve is just velocity times time.

Defining the Area of a Curve in Terms of Riemann Sums

∆x (read as delta x) = width of intervals

Definite Integral Definition

The area of the region bounded by 1. The graph of f 2. The x-axis 3. The line x=a 4. The line x=b Is called the definite integral of f from a to b, and is denoted by b~a f(x)dx or b~a f.

net signed area 1

the area between a function and the x-axis such that the area below the x-axis is subtracted from the area above the x-axis; the result is the same as the definite integral of the function • the quantity A1-A2 is called the net signed area. • net area can be positive, negative, or zero. If the area above the x-axis is larger, the net signed area is positive. If the area below the x-axis is larger, the net signed area is negative. if the areas above and below the x-axis are equal, the net signed area is zero.

Integral Notation

• The integration symbol ∫ is an elongated S, suggesting sigma or summation. • On a definite integral, above and below the summation symbol are boundaries of the interval [a,b]/ The numbers a and b are x-values and are called the limits of integration; a is the lower limit and b is the upper limit. • The function f(x) is the integrand and the dx indicates that f(x) is a function with respect to x, called the variable of integration. • If f(x) is continuous on [a, b], then f is integrable on [a, b].