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Calculating Area Between Curves and the X-Axis Using Definite Integrals

This guide explains how to calculate the area between the x-axis and a continuous function ( f(x) ) over an interval ([a,b]) using definite integrals. The area above the x-axis is treated as positive, while the area below is negative. The process involves setting up integrals for regions bounded by curves ( y = f(x) ) and ( y = g(x) ). We also cover finding area between two curves, determining bounds, and using horizontal and vertical cuts. Examples are provided for integration and area calculation with graphical support.

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Calculating Area Between Curves and the X-Axis Using Definite Integrals

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  1. Area

  2. Back to Area: • We can calculate the area between the x-axis and a continuous function f on the interval [a,b] using the definite integral: • Where f(xi*) is the height of a rectangle and ∆x is the width of that rectangle. {(b-a)/n (n is the number of rectangles)} • Remember that the area above the axis is positive and the area below is negative.

  3. Set up the integral needed to find the area of the region bounded by: and the x-axis.

  4. Set up the integral needed to find the area of the region bounded by: , the x-axis on [0,2].

  5. Area bounded by two curves • Suppose you have 2 curves, y = f(x) and y = g(x) Area under g is: Area under f is:

  6. Superimposing the graphs, we look at the area bounded by the two functions: (top - bottom)*∆x

  7. The area bounded by two functions can be found:

  8. Find the area of the region between the two functions: and Bounds? [-1,2] Top Function? Bottom Function? Area? = 9

  9. Find the area bounded by the curves: and Solve for bounds:

  10. Find the area bounded by the curves: and Sketch the graph: (top - bottom)*∆x

  11. Find the area of the region determined by the curves: and Bounds? In terms of y: [-2,4] Points (-1,-2) & (5,4) Graph? Solve for y:

  12. Find the area of the region determined by the curves: and Need 2 Integrals! One from -3 to -1 and the other from -1 to 5. Area?

  13. Horizontal Cut instead: Bounds? In terms of y: [-2,4] Right Function? Left Function? Area? = 18

  14. In General: Vertical Cut: Horizontal Cut:

  15. Find the Area of the Region bounded by and Bounds? [0,1] Top Function? Bottom Function? Area?

  16. Find the Area of the Region bounded by , , and the y-axis Bounds? [0,π/4] and [π/4, π/2] [0,π/4] Top Function? Bottom Function? Area?

  17. Find the Area of the Region bounded by , , and the y-axis Bounds? [0,π/4] and [π/4, π/2] Top Function? Bottom Function? Area?

  18. Find the area of the Region bounded by Bounds? Interval is from -2, 5 Functions intersect at x = -1 and x = 3 Graph? Top function switches 3 times! This calculation requires 3 integrals!

  19. Find the area of the Region bounded by Area?

  20. Find T so the area between y = x2 and y = T is 1/2. Bounds? Top Function? Bottom Function? Area? Taking advantage of Symmetry Area must equal 1/2: Ans:

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