Areas enclosed by curves and tangent lines, areas of complex regions, and more.
Example ① The area enclosed by y = log x, the x-axis, x = 1, and x = e.
Solution:∫_1^e log x dx = [x log x − x]_1^e = (e − e) − (0 − 1) = 1
Solution:∫_1^e log x dx = [x log x − x]_1^e = (e − e) − (0 − 1) = 1
Key Point
- For definite integrals involving log or e^x, use integration by parts
Practice Problems
- The area enclosed by y = e^x, the x-axis, x=0, and x=1
Answer and Explanation
- Answer:e − 1
∫_0^1 e^x dx = [e^x]_0^1 = e − 1.