Calculation of areas and volumes using definite integral

The area of the curvilinear trapezoid

The area of the curvilinear trapezoid bounded by the graph of the positive continuous on a segment functions, axis and straight and equal

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Example.To calculate the area of the figure bounded by the lines

Portraying these lines produced curvilinear trapezoid

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The area of the figure bounded the graphs of two functions and the direct and

If on a given interval of continuous function and have the property that for all then

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Example.To calculate the area of the figure bounded by the lines

Draw the specified line and the abscissa of their intersection points.

The abscissa of the intersection points:

Then according to the formula

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The volume of the bodies

In the General case

If the body is encased between two perpendicular to the axis planes passing through the points and then

where is the cross-sectional area of the body a plane that passes through the point and perpendicular to the axis

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For the body rotation

If the body obtained by rotation around the axis of a curvilinear trapezoid, limited by schedule and newme continuous functions on the interval and straight and then

Обчислення площ і обємів за допомогою визначеного інтеграла

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