Area Between Two Curves, Integration, With Respect to x and y, Practice Problems - Calculus

The area between two curves, f(x) and g(x), is found by integrating the difference between the upper and lower functions over a given interval [a, b]: Area = ∫[a to b] (f(x) - g(x)) dx. This method works by integrating the area under the upper curve and subtracting the area under the lower curve, similar to finding the area of a square with a circular hole. For functions of y, the formula becomes Area = ∫[y1 to y2] (u(y) - v(y)) dy, where u(y) is the rightmost function and v(y) is the leftmost.

💡Steps to find the area between two curves:
• Identify the functions: Determine the equations of the two curves, often represented as f(x) and g(x).
• Find the limits of integration: Determine the points where the curves intersect to establish the interval [a, b]. If a specific interval is given, use that.
• Determine the upper and lower functions: Identify which function has a greater y-value over the given interval.
• Set up the integral: Write the definite integral of the upper function minus the lower function.
⚬ For integration with respect to x: Area = ∫[a to b] (f(x) - g(x)) dx.
⚬ For integration with respect to y: Area = ∫[y1 to y2] (u(y) - v(y)) dy.
• Evaluate the integral: Calculate the definite integral to find the area. The resulting area will always be a positive value.

💡Example:
To find the area between y = x and y = x² from x=0 to x=1:
• Functions: f(x) = x, g(x) = x².
• Limits: The interval is given as.
• Upper/Lower: For 0 lt x lt 1, x gt x². So, f(x) is the upper function.
• Integral: Area = ∫[0 to 1] (x - x²) dx.
• Evaluate: [1/2 x² - 1/3 x³] from 0 to 1 = (1/2 - 1/3) - (0) = 1/6.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link

💡Chapters:
00:00 Area between curves, functions of x, with example
01:50 Area between curves, functions of y, with example
04:32 Area between curves with intersections, with example

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