\( \int _ { 0 } ^ { 4 } ( x ^ { 3 } - 3 x ^ { 2 } + 1 ) d x \quad n = 6 \)

To evaluate the integral \( \int _ { 0 } ^ { 4 } ( x ^ { 3 } - 3 x ^ { 2 } + 1 ) d x \) with \( n = 6 \), we first need to find the antiderivative of the integrand. The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \), for \( -3x^2 \) it is \( -x^3 \), and for the constant \( 1 \), it is \( x \). Therefore, the antiderivative is given by: \[ F(x) = \frac{x^4}{4} - x^3 + x \] Next, we evaluate this from 0 to 4: \[ F(4) = \frac{4^4}{4} - 4^3 + 4 = \frac{256}{4} - 64 + 4 = 64 - 64 + 4 = 4 \] And now calculate \( F(0) \): \[ F(0) = \frac{0^4}{4} - 0^3 + 0 = 0 \] Now, apply the Fundamental Theorem of Calculus: \[ \int _ { 0 } ^ { 4 } ( x ^ { 3 } - 3 x ^ { 2 } + 1 ) d x = F(4) - F(0) = 4 - 0 = 4 \] Thus, \[ \int _ { 0 } ^ { 4 } ( x ^ { 3 } - 3 x ^ { 2 } + 1 ) d x = 4 \] Now let’s explore some fun facts about integrals and their applications! Did you know that integrals aren’t just for calculus classes? They’re extensively used in fields like physics, engineering, and economics! For instance, when calculating the area under a curve, which could represent everything from the distance traveled by an object to how much profit a business makes over time, integrals do all the heavy lifting. If you ever find yourself confused about setting up your integrals, remember to sketch the function first! This lets your brain visualize what you're integrating over. A common mistake is to misinterpret the bounds of integration, especially if functions have intersections or asymptotic behavior. Visuals can help prevent this, resulting in a more accurate calculation.