Category: Lectures (Page 5 of 9)

Today’s first quiz problem started with an indefinite integral along with specific choices for \(u\) and \(dv\). With \(u = t\), then \(du\) is the derivative: \(du = 1\,dt\). With \(dv = \cos(5t)\,dt\), then \(v\) is antiderivative: \(v = \int \cos(5t)\,dt\).

Today’s first quiz problem is a definite integral. It is important to consider what our final result should look like — in this case, as a definite integral, our result should be a real number (representing the area bounded by \(f(x)\), the \(x\)-bounds, and the \(x\)-axis).

The first quiz problem is a definite integral. We started by thinking about what the function \(f(x) = \sqrt[4]{x}\) looks like when graphed, with the bounds \(x=1\) and \(x=16\). This region is not a “nice” geometric shape — so computing the area will require the Fundamental Theorem of Calculus.