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Integration can be used to calculate areas. In simple cases, the area is given by a single definite integral. But sometimes the integral gives a negative answer which is minus the area, and in more complicated cases the correct answer can be obtained only by splitting the area into several parts and adding or subtracting the appropriate integrals.
Integration by Parts. Name___________________________________. Date________________ Period____. Evaluate each indefinite integral using integration by parts. u and dv are provided. 1) ∫ x x e dx; u = x, dv = x e dx. x x x e − e + C. 3) ∫ x ⋅ 2 x dx; u = x, dv = 2 dx. x.
Express f(x) in the form a(x + b)2 + c, stating the values of the constants a, b and c. State the coordinates of the turning point of the curve y = f(x). Find the area of the region enclosed by the curve y = f(x) and the line y = 3. a Evaluate. 2 8.
There are two ways to solve this problem: we can calculate the area between two functions and using the vertical elements and integrate with respect to x, or we can use the horizontal elements and calculate the area between the y -axis and the function integrating the functions with respect to y.
Use calculus to find the total area of the finite region, shown shaded in the diagram, that is between
Use your expression in a and integration to find an estimate for the area of R. Use the trapezium rule with 6 equally-spaced ordinates to find another estimate for the area of R. 8= x2 + 16xR4 xThe diagram shows the curve. = x2 + 16 for x > 0.Show that the stationary point on the curve.
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