Methods Of Integration

Step 1: Identify the intersection points
The given equations are: Equating both, Solving for , For , substituting in , Thus, .
Step 2: Compute the Area
The required area is obtained using the formula: After performing the necessary integration and simplifications,
Step 3: Final Answer
Thus, the area enclosed by the given curves is:

Step 1: Solve the Integral
We solve: Using the standard substitution: Since, We get, Thus,
Step 2: Find
Substituting :
Step 3: Conclusion
Thus, the value of is:

Step 1: Identify the Breakpoint
The given function is , which changes its definition when . Thus, we split the integral at .
Step 2: Split the Integral into Two Parts

Step 3: Evaluate the First Integral
Evaluating at : Evaluating at : Thus,
Step 4: Evaluate the Second Integral
Evaluating at : Evaluating at : Thus,
Step 5: Compute the Final Result
Thus, the final result is:

Step 1: Substituting the given limits and simplifying the integral expression
Given the integral, Rewriting the terms in a simpler form, Using standard substitution techniques, we analyze the function's structure and solve the integral.
Step 2: Evaluating the Integral
After evaluating the given integral using appropriate transformations and approximations,
Step 3: Final Answer
Thus, the computed value of the given integral is: