Calculating Area Between Two Curves using Calculus
Learn how to calculate the area between two curves using calculus in mathematics. Understand the concept of finding areas of irregular shapes and the method to find the area between a curve and a line. Explore examples to grasp the calculations involved.
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12 June 2025 Area between two curves LO: Use calculus to calculate the area between two curves www.mathssupport.org
Areas of irregular shapes We have already stablished that: If f is a positive and continuous function on the interval a x b, then the area bounded by y = f(x), the x-axis and the vertical lines x = a and x = b is given by y x 0 a b ? ? ? ?? ? www.mathssupport.org
The area between a curve and a line Suppose we want to find the area between a curve and a line. Find the area bounded by the line f (x) and the curve g (x). We can get an approximation of the area using parallelograms The height of each parallelogram Top graph bottom graph fi(x) gi(x) y f(x) g(x) f1(x) The width of each parallelogram is dx g1(x) dx The area of each parallelogram is (fi(x) gi(x)) dx The total area is ? (? ? ? ? )?? a b x As the number of parallelograms increases, the approximate area approaches the actual area ? www.mathssupport.org
The area between two curves The method used to find the area between a curve and a line can also be used to find the area between two curves. In general: y y = f(x) The area A between two curves y = f(x) and y = g(x) is given by ( = ( ) a A y = g(x) b ) ( ) g x dx A f x x a b www.mathssupport.org
The area between a curve and a line Suppose we want to find the area between a curve and a line. For example: Find the area bounded by the curve y = x2 + 2 and the line y = 2x + 5. Start by sketching the graph, and shading the required area. Next, we have to find the x-coordinates of the points where the curve and the line intersect. y f(x) = x2 + 2 x2 + 2 = 2x + 5 x2 2x 3 = 0 (x 3)(x + 1) = 0 Solving: g(x) = 2x + 5 -1 3 x the curve and the line intersect when x = 1 and x = 3. www.mathssupport.org
The area between a curve and a line The area under the line y = 2x + 5 between x = 1 and x = 3 is this area: While the area under the curve y = x2 + 2 between x = 1 and x = 3 is this area: y y = x2+ 2 So the area we require, A, can be found by subtracting the area under the curve from the area under the line. A 1 3 x y = 2x + 5 This area is given by: 3 3 ?2+ 2 ?? ? = 2? + 5 ?? 1 1 www.mathssupport.org
The area between a curve and a line Since the limits are the same these two integrals can be combined to give 3 2? + 5 ?2+ 2 ?? ? = 1 3 ?2+ 2? + 3 ?? = 1 3 = 1 3?3+ ?2+ 3? = ( 9 + 9 + 9) (1 = 9 + 11 3 = 101 3 1 3+ 1 3) 1 3 So the required area is 10 units2. www.mathssupport.org
The area between two curves Find the area bound by the curves f(x) = 2e ? 2 and g(x) = x2 4x. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection MENU 6 Type in Y1 = 2e ? 2 Y2 = x2 4x EXE F6 y f(x) = 2? ? ? 2 = x2 4x x A g(x) = x2 4x www.mathssupport.org
The area between two curves Find the area bound by the curves f(x) = 2e ? 2 and g(x) = x2 4x. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection y f(x) = 2? ? ? 2 = x2 4x x A g(x) = x2 4x EXE F6 DRAW www.mathssupport.org
The area between two curves Find the area bound by the curves f(x) = 2e ? 2 and g(x) = x2 4x. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection y f(x) = 2? ? ? 2 = x2 4x x A g(x) = x2 4x F5 G-Solv INTSECT F5 www.mathssupport.org
The area between two curves Find the area bound by the curves y = 3 2x2 and x = 1 x2. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection y f(x) = 2? ? ? 2 = x2 4x x A f(x) = x2 4x x 0.5843 www.mathssupport.org
The area between two curves Find the area bound by the curves y = 3 2x2 and x = 1 x2. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection EXIT MENU 1 F4 MATH F6 y f(x) = 2? ? ? 2 = x2 4x x A f(x) = x2 4x x 0.5843 x 4.064 ?? F1 www.mathssupport.org
The area between two curves Find the area bound by the curves y = 3 2x2 and x = 1 x2. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection y f(x) = 2? ? ? 2 = x2 4x x A f(x) = x2 4x EXE x 0.5843 x 4.064 Type in the integral: 4.064 2? ? 2 ?2 4? ???? = ?? 0.5843 www.mathssupport.org
The area between two curves Find the area bound by the curves y = 3 2x2 and x = 1 x2. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection y f(x) = 2? ? ? 2 = x2 4x x A f(x) = x2 4x x 0.5843 x 4.064 Simplify the integral: 4.064 2? ? 2 ?2 4? ???? = ?? 0.5843 www.mathssupport.org
The area between two curves Find the area bound by the curves y = 3 2x2 and x = 1 x2. Start by equating the curves to find out where they intersect. 2e ? Use the GDC to sketch the graphs and find the x-coordinate of the points of intersection y f(x) = 2? ? ? 2 = x2 4x x A f(x) = x2 4x x 0.5843 x 4.064 Simplify the integral: 4.064 2? ? 2 ?2 4? 14.7 ???? = ?? 0.5843 www.mathssupport.org
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