Let f: [0, ∞) →R be a differentiable function such that f(x) = 1 - 2x + ∫0xet−xf(t) \, dt for all x ∈[0,∞). Then the area of the region bounded by y=f(x) and the coordinate axes is:
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Step-by-Step Solution
Step 1: Rewrite the integral equation
The given integral equation involves et−x. We can rewrite et−x as ete−x. Since e−x is independent of the integration variable t, we can take it out of the integral. This simplifies the expression and prepares it for differentiation.
Step 2: Differentiate the equation
We differentiate both sides of the rewritten equation with respect to x. For the integral term, we use the product rule for differentiation, treating e−x as one function and ∫0xetf(t)dt as another. The derivative of the integral part is found using the Leibniz integral rule, which states that dxd∫a(x)b(x)g(x,t)dt=g(x,b(x))b′(x)−g(x,a(x))a′(x)+∫a(x)b(x)∂x∂g(x,t)dt. In our case, a(x)=0, b(x)=x, and g(x,t)=etf(t), so the derivative is exf(x).
Step 3: Simplify and form a differential equation
We substitute the original expression for e−x∫0xetf(t)dt back into the differentiated equation. This allows us to eliminate the integral term and simplify the equation into a first-order linear differential equation. The terms involving f(x) cancel out.
Step 4: Integrate to find f(x)
We integrate f'(x) with respect to x to find f(x). This yields a general solution with an integration constant C. To find C, we use the initial condition f(0). From the original equation, substituting x=0 gives f(0)=1−2(0)+∫00et−0f(t)dt=1. Therefore, f(0)=1. Substituting x=0 into our integrated function gives f(0)=−0−02+C=C. So, C=1.
Step 5: Determine the function f(x)
With the constant of integration C=1, the function f(x) is determined as a quadratic polynomial. This function defines the curve whose area we need to find.
Step 6: Find x-intercepts and set up the integral for area
To find the area bounded by y=f(x) and the coordinate axes, we first need to find the x -intercepts of f(x) by setting f(x)=0. Using the quadratic formula, we find two roots. Since the domain of f(x) is [0,∞), we are interested in the positive root, which is x=2−1+5. The y -intercept is f(0)=1. The area will be the integral of f(x) from 0 to this positive x -intercept.
Step 7: Calculate the area
We integrate f(x) from 0 to the positive x -intercept x0=25−1. We evaluate the definite integral and substitute the limits. After careful calculation, the area is found to be 2455−1.