Question

Find the area under the curve f(x) = |1-x⁴| from limits 1 to 3.

Answer 1

Answer:

232/5 unit²

Step-by-step explanation:

     f(x)   = 1 - x⁴       , 1 - x⁴ > 0

             = -(1 - x⁴)    , 1 - x⁴ < 0

But for the given limit f(x) is always negative(i.e. 1 - x⁴ < 0)

   ∴ f(x) = - (1 - x⁴)  = x⁴ - 1

As f(x) is now positive for x in given limits, we don't need to care about any area below x-axis (-ve).  Therefore we can say

        Area under graph = $\mathsf{\int\limits^3_1 {(x^4 - 1)} \, dx }$

               = $\mathsf{\bigg(\dfrac{x^5}{5} - x\bigg) \bigg|^3_1}$

               = $\mathsf{\dfrac{232}{5} }$