Visualizing ∫(4+5√x)^2dx with Animations

To expand this squared binomial, we'll multiply (4 + 5√x) by itself using the FOIL method.Let's write out the expression with both brackets.First, we multiply the constant terms: 4 times 4 equals 16.Next, we multiply the outer and inner terms: 4 times 5√x, and 5√x times 4. These are like terms that combine to 40√x.Finally, we multiply the last terms: 5√x times 5√x. Since √x times √x equals x, this gives us 25x.Now we can combine all these terms into our expanded expression.Each term in our expansion comes from a different part of the FOIL process.Now we'll integrate each term separately using the power rule.For our first term, integrating sixteen d x, we treat sixteen as a constant.For forty square root of x, we first convert the square root to a fractional exponent.The square root becomes x to the one-half power. When we integrate, we add one to the exponent and divide by the new power.Finally, for twenty-five x, we apply the power rule directly.The power of x increases from one to two, and we divide by the new power.Now we'll combine our integrated terms and simplify the expression.Let's simplify the coefficients. For the x to the three-halves term, forty times two-thirds becomes eighty-thirds.For the x-squared term, twenty-five times one-half becomes twenty-five halves.This gives us our simplified expression.To understand how each term contributes to the total area, let's visualize their growth on a graph.The linear term, sixteen x, grows at a constant rate.The x to the three-halves term grows faster than linear, but slower than quadratic.The quadratic term, twenty-five halves x squared, eventually dominates as x increases.As we move along the x-axis, notice how the relative contributions of each term change.