Evaluate
by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.
Evaluate what?
To evaluate the expression
∫(x^2 + 1) dx
as a sum of two integrals, we can split it into two separate integrals as follows:
∫x^2 dx + ∫1 dx
The first integral, ∫x^2 dx, is a typical integral of a polynomial function. To evaluate it, we can use the power rule of integration, which states that ∫x^n dx = (1/(n+1)) x^(n+1) + C. Applying this rule, we have:
∫x^2 dx = (1/3) x^3 + C1
where C1 is the constant of integration.
The second integral, ∫1 dx, is a bit simpler. Since the derivative of a constant is zero, integrating a constant is simply multiplying it by x. Therefore:
∫1 dx = x + C2
where C2 is another constant of integration.
Thus, the expression
∫(x^2 + 1) dx
can be rewritten as:
(1/3) x^3 + x + C
where C = (C1 + C2) is the combined constant of integration.
Interpreting one of those integrals in terms of an area:
The second integral, ∫1 dx, can be interpreted in terms of an area. Since we are integrating the constant function 1, the area under the curve is equal to the width of the interval of integration.
For example, if we are integrating over the interval [a, b], the area under the curve would be:
∫[a, b] 1 dx = (b - a)
This is because the value of the integral is equal to the value of the function, which is 1, multiplied by the width of the interval ([b - a]).
So, in the expression
∫(x^2 + 1) dx
the second integral, ∫1 dx, represents the area under the curve of the constant function 1 over the interval of integration.