Problem 1 based on Application of Definite Integration (Area bounded by two Curves)

Area Under a Curve: - The area between the curve y = f(x), the x-axis, and the lines x = a and x = b is given by the definite integral: A = Integration a to b f(x) dx
If f(x) is greater than equal to 0, the area is positive.
If f(x) is less than equal to 0, the area is negative, and you take the absolute value for physical interpretation.
Area Between Two Curves: - If f(x) and g(x) are two continuous functions where f(x) is greater than equal to g(x) in the interval [a, b], the area between the curves y = f(x) and y = g(x) is given by: A = integration a to b [upper curve – lower curve] dx = integration a to b [f(x) – g(x)] dx
Area in Parametric Form: - If the curve is expressed in parametric form x = f(t) and y = g(t), where t lies in the interval [t1, t2], the area enclosed is:
A = integration t1 to t2 g(t) dx/dt dt = integration t1to t2 g(t) f'(t)dt
Area in Polar Coordinates: - For a curve given in polar coordinates as r = f(theta), the area enclosed between angles (theta1) and (theta2) is:
A = 1/2 integration {theta1} to {theta2} [f(theta)]^2 d(theta)
Properties of Definite Integrals:
Reversal of Limits: integration a to b f(x) dx = -integration b to a f(x), dx
Additivity: integration a to b f(x)dx = int_a to c f(x)dx + integration c to b f(x)dx
Zero Integral: integration a to a f(x)dx = 0
Even and Odd Functions: For an even function f(x) = f(-x), the integral from -a to a is: integration –a to a f(x)dx = 2 integration 0 to a f(x)dx
For an odd function f(x) = -f(-x), the integral from -a to a is: integration –a to a f(x)dx = 0
Volume of Solids of Revolution: - The volume of a solid obtained by revolving a region around the x-axis (using the disc method) is given by: V = pi integration a to b [f(x)]^2dx
The volume of a solid obtained by revolving a region around the y-axis is given by: V = pi integration a to b [g(y)]^2dy

Application of Definite Integration" and "Area of Curves" in Class 12
Definite Integrals
Area Under a Curve: Compute the area between a curve and the x-axis.
Area Between Two Curves: Methods to find the area enclosed between two functions.
Real-world applications of definite integrals in physics problems.
Integration Techniques: Various methods of integration used in solving area-related problems.
Area Between a Curve and the y-Axis
Area as Definite Integral
Using Definite Integrals to Find Area and Length
Areas Between Curves
Detailed explanations and examples on calculating the area between two curves using definite integrals.
Finding The Area Under The Curve Using Definite Integrals:
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