Area bounded by curves using integration: The area bounded by curves using integration is the total area of the region bounded by two curves. We can fin...
Area bounded by curves using integration:
The area bounded by curves using integration is the total area of the region bounded by two curves. We can find this by subtracting the area of the region below the curve from the area of the region above the curve.
How to find the area:
Identify the curves: We need to identify the two curves that bound the region.
Find the points of intersection: Determine the points where the two curves intersect.
Determine the function range: For each curve, determine the range of values the independent variable (usually x) can take on.
Set up the integral: Integrate the function representing the upper curve (y = f(x)) minus the function representing the lower curve (y = g(x)) with respect to the independent variable.
Evaluate the integral: Calculate the total area by evaluating the definite integral.
Examples:
Area of a region between two curves: If f(x) = x^2 and g(x) = x, the area bounded by these curves between x = 0 and x = 2 is found by integrating f(x) - g(x) dx from 0 to 2.
Area of a region bounded by two circles: The area bounded by the circles centered at the origin with radius 2 and 3 is found by finding the difference between the areas of the circles using the formula for the area of a circle.
Area of a region bounded by a curve and a line: The area bounded by the curve y = x^2 and the line y = 0 when x is between 0 and 2 is found by integrating x^2 from 0 to 2 with respect to x