Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental Theorem of Calculus

Applying the well known formula: Distance Traveled A train moves along a track at a steady rate of 75 miles per hour from 3:00 A.M. to 8:00 A.M. What is the total distance traveled by the train? Applying the well known formula: Velocity (mph) Notice the distance traveled by the train (375 miles) is exactly the area of the rectangle whose base is the time interval [3,8] and whose height is the constant velocity function v=75. Time (h)

Distance Traveled A particle moves along the x-axis with velocity v(t)=-t2+2t+5 for time t≥0 seconds. How far is the particle after 3 seconds? Velocity The distance traveled is still the area under the curve. Unfortunately the shape is a irregular region. We need to find a method to find this area. Time

The Area Problem We now investigate how to solve the area problem: Find the area of the region S that lies under the curve y=f(x) from a to b. f(x) This means S is bounded by the graph of a continuous function, two vertical lines, and the x-axis. S a b

Finding Area It is easy to calculate the area of certain shapes because familiar formulas exist: A=½bh A=lw The area of irregular polygons can be found by dividing them into convenient shapes and their areas: A2 A3 A1 A4

Approximating the Area Under a Curve We first approximate the area under a function by rectangles. a b

Approximating the Area Under a Curve Then we take the limit of the areas of these rectangles as we increase the number of rectangles. a b

Approximating the Area Under a Curve Then we take the limit of the areas of these rectangles as we increase the number of rectangles. a b

Estimating Area Using Rectangles and Right Endpoints Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and right endpoints. Make rectangles whose base is the same as the strip and whose height is the same as the right edge of the strip. Width = ¼ and height = value of the function at ¼ Find the Sum of the Areas: Divide the area under the curve into 4 equal strips

Estimating Area Using Rectangles and Right Endpoints Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and right endpoints. Make rectangles whose base is the same as the strip and whose height is the same as the right edge of the strip. Width = 1/8 and height = value of the function at 1/8 Find the Sum of the Areas: Divide the area under the curve into 8 equal strips

Estimating Area Using Rectangles and Left Endpoints Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and left endpoints. Make rectangles whose base is the same as the strip and whose height is the same as the left edge of the strip. Width = ¼ and height = value of the function at 0 Find the Sum of the Areas: Divide the area under the curve into 4 equal strips

Estimating Area Using Rectangles and Left Endpoints Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and left endpoints. Make rectangles whose base is the same as the strip and whose height is the same as the left edge of the strip. Width = 1/8 and height = value of the function at 0 Find the Sum of the Areas: Divide the area under the curve into 8 equal strips

Distance Traveled A particle moves along the x-axis with velocity v(t)=-t2+2t+5 for time t≥0 seconds. Use three midpoint rectangles to estimate how far the particle traveled after 3 seconds? Width = 1 and height = value of the function at 0.5 Make rectangles whose base is the same as the strip and whose height is the same as the middle of the strip. Velocity Find the Sum of the Areas: Divide the area under the curve into 3 equal strips Time

Negative Area If a function is less than zero for an interval, the region between the graph and the x-axis represents negative area. Positive Area Negative Area

Definite Integral: Area Under a Curve If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b. Upper limit of integration Lower limit of integration

The Existence of Definite Integrals All continuous functions are integrable. That is, if a function f is continuous on an interval [a,b], then its definite integral over [a,b] exists . Ex:

Rules for Definite Integrals Let f and g be functions and x a variable; a, b, c, and k be constant. Constant Multiple Sum Rule Difference Rule Additivity

The First Fundamental Theorem of Calculus If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then

Example 1 Evaluate First Find the indefinte integral F(x): Now apply the FTC to find the definite integral: Notice that it is not necessary to include the “C” with definite integrals

Examples: New Notation 1. Evaluate F(x) Bounds 2. Evaluate 3. Evaluate