91Ó°ÊÓ

The magnitude of a vector gives us a sense of its "length" in space. In this context, we're dealing with a vector function \( \mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} \). The magnitude of this vector \( \| \mathbf{r}(t) \| \) is the length from the origin to the point \( (t, t^2) \) in the plane. It's computed using the formula for Euclidean distance in vector terms, which is: \( \sqrt{(t^2) + (t^2)^2} = \sqrt{t^2 + t^4} \). This will be used in the integral to find the total "length" or "magnitude" of the vector function from \( t = 0 \) to \( t = 2 \). Understanding vector magnitude is crucial as it translates a multidimensional function into a single dimension that we can work with in calculus.

Integration by substitution is a method used to simplify complex integrals by changing variables. It works by choosing a substitution that transforms an integral into a simpler form. In this example, to evaluate \( \int_{0}^{2} t \sqrt{1 + t^2} \, dt \), we use substitution. We let \( u = 1 + t^2 \). This means \( du = 2t \, dt \), allowing us to replace \( t \, dt \) with \( \frac{1}{2} du \). This changes the integral into a simpler form: \( \int_{1}^{5} \frac{1}{2} \sqrt{u} \, du \). By simplifying the integrand, we can manage the calculus steps more easily. Substitution effectively transforms the problem into a more familiar and straightforward integral to solve.

When we change variables in an integral via substitution, we also need to change the limits of integration to reflect the new variable. Originally our limits were in terms of \( t \), from \( 0 \) to \( 2 \). After substitution with \( u = 1 + t^2 \), the limits must also switch. For \( t = 0 \), we calculate \( u = 1 + 0^2 = 1 \). For \( t = 2 \), \( u = 1 + 2^2 = 5 \). This transformation of limits ensures that when you evaluate the new integral \( \int_{1}^{5} \frac{1}{2} \sqrt{u} \, du \), you are calculating the same area or total as the original integral in a different form. It's crucial not to forget this step, as using the original limits with the new variable would yield incorrect results.

The calculus evaluation steps involve breaking down the solution to an integral step by step. In our example, we start by simplifying the integrand using algebraic manipulation: \( \sqrt{t^2 + t^4} = t \sqrt{1 + t^2} \). Next, we apply substitution to transform the integral into one that’s easier: \( \int_{1}^{5} \frac{1}{2} \sqrt{u} \, du \). The integration result is \( \int \frac{1}{2} u^{1/2} \, du \), calculated as \( \frac{1}{2} \left[ \frac{u^{3/2}}{3/2} \right] \). After simplifying, it becomes \( \frac{1}{3} [u^{3/2}] \), which is then evaluated between the new limits, 1 and 5. This step by step process ensures that each part of the problem is manageable and the solution is reached systematically. Finally, we compute the definite integral to find the result, \( \frac{1}{3}(5\sqrt{5}) - \frac{1}{3} \), leading to a tidy answer: \( \frac{5\sqrt{5} - 1}{3} \). These steps showcase the methodical approach needed in calculus to solve problems accurately.