Question

Show that the value of \(\int _ { 0 } ^ { 1 } \sin \left( x ^ { 2 } \right) d x\) cannot possibly be 2

Short answer

No, the value of \(\int _ { 0 } ^ { 1 } \sin \left( x ^ { 2 } \right) dx\) cannot possibly be 2 because the maximum possible value for the integral of any function going from 0 to 1 can be 1.

Step-by-step solution

Examine the Function

First, examine the function we are integrating, \(\sin \left( x ^ { 2 } \right)\). Note that the sine function oscillates between -1 and 1 for all real values. So, the values of \(\sin \left( x ^ { 2 } \right)\) will also lie between -1 and 1.

Observe the Area Under the Curve

The process of integration is a way to find the total area under the curve of a function. With this particular function, none of the areas below the x - axis will be included in the integral, because it goes from 0 to 1. That is, all values of x will be greater than or equal to zero, which means none of the areas below the x-axis will count towards our integral.

Check Maximum Possible Value

Given that the function we are integrating, \(\sin \left( x ^ { 2 } \right)\), will always be less than or equal to 1 for 0 ≤ x ≤ 1, the largest possible value for the definite integral from 0 to 1 of \(\sin \left( x ^ { 2 } \right) dx\) would be the integral of 1 dx from 0 to 1, which equals to 1.

Conclusion

Since the maximum possible value for the integral is 1, it cannot possibly be 2.