Calculus Volume 1 · 免费预览
曲线间的面积
由两条曲线围成区域的面积，等于上方曲线减去下方曲线的积分；若用水平条带切片，则为右边界减去左边界的积分，交点则确定积分限。
1. Vertical slices
Vertical slices
2. Horizontal slices
Horizontal slices
3. Curves that cross
Curves that cross
4. Areas with geometric meaning
Areas with geometric meaning
5. Top minus bottom setup
The region between f(x) = 2 - x^2 and g(x) = x on [-2, 1] is sliced vertically. What integrand gives the strip height?
6. Finding limits from intersections
To find the area between two curves, the limits of integration are usually determined by what?
7. Horizontal slicing integrand
A region is sliced horizontally, bounded on the right by x = h(y) and on the left by x = k(y). What is the strip length?
8. Splitting at a crossing
Two curves cross at x = c inside [a, b], exchanging top and bottom positions. What must the area integral do?
9. Equal area by either slicing
The region between y = x and y = x^2 was computed as 1/6 by vertical slices and 1/6 by horizontal slices. What does the agreement demonstrate?
10. Which curve is on top
For the region between y = 2 - x^2 and y = x at the test point x = 0, which curve lies on top?
11. Area of a simple gap
The area between y = x and y = x^2 from 0 to 1 equals the integral of (x - x^2) dx. What fraction is it? Answer like 1/6.
12. Integrand sign audit
The area between y = 4 and y = x^2 on [-2, 2] has strip height 4 - x^2, always nonnegative there. At x = 0, what value does the integrand take? Answer with a number.
13. Intersection count
The curves y = 2 - x^2 and y = x intersect where x^2 + x - 2 = 0. How many intersection points enclose the finite region? Answer with a number.
14. Horizontal slice dy
When slicing horizontally, the integration variable is y and the differential is dy. As a two-letter string, what is that differential? Answer in lowercase.