We have seen that the area under the graph of a function y / () and above the x-axis can be approximated by using a series of calculated areas of rectangular regions. Using more rectangles will yield a more accurate approximation. In fact, we found that the exact area could be compute by finding the limit of an infinite number of rectangular regions. lim tn1 Here n is the number of rectangles, fis a function evaluated at an x-value in each rectangle, and Ax is the width of the rectangles. If n is a finite number then we have only computed an approximate area. By determining the limit as n goes to infinity, we find the exact area. We now wish to develop a simple means of finding that same area using another calculus-based method, a principal known as the Fundamental Theorem of Calculus. Development We already found the area bounded by the function f(t) = 2t + 3 on the interval from t=1 and t 4 using geometrical concepts. f() - 2t +3 14 12 10 Find the area of the shaded region using the formula for the area of a trapezoid, A = (b; + b2). %3D 1. 1 to t= x. Simplify this expression. 2. Using the same formula, calculate the area bounded by the graph of f (t) from t Your anwer from part (2) can be thought of as an area function, Ale), which measures the area under the graph from r1 totex Find A(4) using this function. How does this value compare to your calculation in part (1)7 4. Use the area function Ax) to find A(1). What does this represent geometrically? Use the area function Alx) to find the area from x3 to x6. 5 6. Use the area function Alx) to find the area from O to x=3. 7. How can we determine an expression for the rate of change of this function A (W? Find this expression. 8. What is the relationship between the area function A(x) and the original function in our graph, f(t)? 9. Use the area function Alx) to find the area from x=1 to x a. 10. Use the area function A(x) to find the area from x=1 to x -b. 11. Assuming that a