math1014_ch6_filled - MATH 1014 Calculus II Chapter 6 Applications of Integration 1 Areas between curves Suppose that the curve of y = f(x is above the

# math1014_ch6_filled - MATH 1014 Calculus II Chapter 6...

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MATH 1014 - Calculus II Chapter 6 - Applications of Integration 1. Areas between curvesSuppose that the curve ofy=f(x)is above thex-axis(i.e.,f(x)0)in the intervalx[a, b]. We 1
Depending on the geometry, sometimes it may be more convenient to evaluate the area of the regionbounded between two curves usingyas the independent variable.In below, the area of the regionbounded between the curvesx=f(y)andx=g(y), and the linesx=candx=dcan be given by thedefinite integral (forf(y)g(y))A=´dc[f(y)-g(y)]dy= limn→∞nk=1[fyk)-gyk)]triangley.The Riemann sun is to sum up the areas of the rectangles with fixed heighttriangleyand variable widthsfyk)-gyk).E.g. 3. Find the area bounded the curvesx=y2-4andy=x/3.[ans: 125/6] x= 2-y2andx=|y|.[ans: 7/3] 2
2. Evaluating volume by slicing approachIf the cross-sectional areaAof a solid body along thex-axiscan be described analytically by a single-variable function, i.e.,A(x), we may evaluate the volume of the body by the approach of slicing alongthex-axis. Let the dimension of the solid extends fromx=atox=b, and the cross-section area isgiven by the functionA(x). We divide the interval[a, b]intonsub-intervals so that each has the lengthtrianglex=b-an. When the solid is cut intoninfinitely thin slices (n→ ∞)at these sub-intervals each slicethen has the thickness oftrianglex, and so in thek-th slice (k= 1, . . . , n) the cross-sectional areaAcan beassumed to be constant within the sub-interval, sayAxk), where¯xkis thexcoordinate of a point withinthek-th sub-interval. The volume of thek-th sub-interval is thentriangleVk=Axk)trianglex. The volumeVof