Question:

If $\int^{2}_{-3} f\left(x\right)dx = \frac{7}{3} $ and $\int^{9}_{-3} f\left(x\right)dx = - \frac{5}{6} , $ then what is the value of $\int^{9}_{2} f\left(x\right)dx $ ?

Updated On: Jul 6, 2022
  • $ - \frac{19}{6}$
  • $ \frac{19}{6}$
  • $\frac{3}{2}$
  • $ - \frac{3}{2}$
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Solution and Explanation

Value of the integral $\int^{9}_2 f(x) dx$ $= \int^{9}_{-3} f\left(x\right)dx - \int^{2}_{-3} f\left(x\right)dx $ Given, $\int^{9}_{-3} f\left(x\right)dx = \frac{-5}{6}$ and $ \int^{2}_{- 3}f\left(x\right)dx = \frac{7}{3} $ Putting these values in equation (i) $\int^{9}_{2} f\left(x\right)dx = \frac{-5}{6} - \frac{7}{3} = - \frac{19}{6} $
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Notes on integral

Concepts Used:

Integral

The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.

  • The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
  • The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
  • An integral of a function over an interval on which the integral is described.

Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.

F'(x) = f(x)

For every value of x = I.

Types of Integrals:

Integral calculus helps to resolve two major types of problems:

  1. The problem of getting a function if its derivative is given.
  2. The problem of getting the area bounded by the graph of a function under given situations.