The area between two curves represents the region bounded by their functions. To find this area, we integrate the difference between the upper and lower functions over the specified interval.
The process involves identifying where the curves intersect (to determine integration bounds), determining which function is on top, and setting up the integral: \[\int_a^b [f(x) - g(x)]dx\]
When the curves cross, we may need to split the integral at intersection points and consider which function is above or below in each region. Sometimes, it's more convenient to integrate with respect to y instead of x, especially when the curves are better expressed as functions of y.
Sometimes integrating with respect to y is more natural or simpler than integrating with respect to x. This requires rewriting our integral using \(dy\) instead of \(dx\). The bounds of integration will change to reflect y-values instead of x-values.
When changing variables, we must carefully consider how this affects our integral setup. The relationship between differentials (\(dx\) and \(dy\)) becomes crucial. For example, if \(y = x^2\), then \(dy = 2x\,dx\), and we can use this to transform our integral.
This technique is particularly useful when dealing with curves that are better expressed as functions of y, or when the region we're measuring is more naturally bounded by horizontal lines rather than vertical ones.
Cross-sectional area problems involve finding the volume of a solid where we know the area of each cross-section perpendicular to an axis. The key is to express the cross-sectional area as a function of position along the axis of integration.
The volume formula becomes \[V = \int A(x)\,dx\] where \(A(x)\) represents the area of the cross-section at position x. Common cross-sectional shapes include squares, rectangles, triangles, and circles, each with their own area formulas.
For example, if a solid has circular cross-sections, \(A(x) = \pi r(x)^2\), where \(r(x)\) is the radius at position x. The challenge often lies in determining how the dimensions of the cross-section relate to the position along the axis.
The disc method finds the volume of a solid formed by rotating a region around an axis. Each cross-section perpendicular to the axis of rotation is a circular disc. The volume is found using \[V = \pi\int [f(x)]^2\,dx\] where \(f(x)\) represents the radius of each disc.
The washer method extends this concept to find the volume when we rotate a region between two curves, creating circular washers instead of solid discs. The volume formula becomes \[V = \pi\int [R(x)^2 - r(x)^2]\,dx\] where \(R(x)\) and \(r(x)\) are the outer and inner radii.
Key considerations include choosing the appropriate axis of rotation, determining the bounds of integration, and correctly identifying the functions that generate the inner and outer radii. The method can be adapted for rotation around any horizontal or vertical line, not just the coordinate axes.
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