Sigma Notation and Riemann Sums

When finding the area under a curve, we first need to understand how to add up many small pieces. That's where sigma notation comes in! The symbol ∑ (sigma) represents a sum, with the numbers above and below telling you where to start and end.

A Riemann sum is what we call the area approximation found by adding up rectangles under a curve. The interval we're looking at is divided into pieces called subintervals, and all these subintervals together form the partition. These subintervals don't need to be the same size!

The accuracy of our approximation depends on how small our rectangles are. As the norm of the partition (length of the longest subinterval) approaches zero, our approximation gets better and better. This leads to the formula: Area = lim‖P‖→0 ∑f(ck)Δxk

💡 Think of it this way: the smaller you make each rectangle, the less empty or overlapping space you'll have, giving you a more accurate area calculation!

The Definite Integral

The definite integral represents the exact area under a curve, found by taking the limit of Riemann sums as we make infinite rectangles. For a function on an interval [a,b], we write this as:

∫[a to b] f(x)dx

The good news? If a function is continuous on an interval, its definite integral definitely exists! This means for continuous functions, we can use regular partitions where all subintervals have the same length.

The notation has specific parts: the integration symbol ∫, lower and upper limits (a and b), the function f(x) called the integrand, and dx which represents the infinitely small width of each rectangle. This elegant notation was introduced by Leibniz and makes complex calculations much clearer.

When you see ∫[a to b] f(x)dx, you're looking at the exact area under the curve from point a to point b. The definite integral transforms our approximation into a precise value.

🔍 Notice that the variable x in the integral is just a placeholder (called a dummy variable). You could use any letter (t, u, etc.) and the integral would be the same value!

Calculating Areas and Applications

You can express limits as definite integrals when they involve sums with infinitely small partitions. For example, lim[n→∞] ∑$k=1 to n$ ck² Δx over [0,2] becomes ∫[0 to 2] x²dx.

For a non-negative function, the area under the curve is exactly the definite integral of that function. This gives us a powerful tool for calculating quantities in the real world.

Consider finding distance traveled: if you know an object's velocity function is v = ½t + 1, the distance traveled after 4 seconds is ∫[0 to 4] $½t + 1$dt. Since this is a linear function (forming a trapezoid), you can use geometry instead of calculus: A = ½(1+3)(4) = 8 units.

🚗 Distance traveled equals the area under the velocity curve! This connection between calculus and physics helps you understand how calculus applies to the real world.

Signed Areas and Special Cases

Integrals don't just measure areas—they measure signed areas. When a function dips below the x-axis, those areas count as negative. This means the definite integral equals the area above the x-axis minus the area below it.

This concept helps you evaluate many integrals by thinking about geometric properties. For example, knowing that ∫[0 to π] sin x dx = 2, you can figure out related integrals like ∫[0 to 2π] sin x dx = 0 because the positive and negative areas cancel out.

You can also use properties of functions to evaluate integrals:

• Stretching functions (like multiplying by 2) stretches the area
• Shifting functions $like sin(x-2)$ shifts where the area is measured
• Changing the integration variable (like using u instead of x) doesn't change the value

⚖️ Remember that for odd functions like sin(x), the integral from -a to a equals zero because the positive and negative areas perfectly balance each other out!

Constant Functions and Discontinuities

When a function is constant $f(x) = c$ over an interval [a,b], calculating its definite integral is simple: ∫[a to b] c dx = c$b-a$. This makes sense visually—it's just the area of a rectangle!

For example, a train moving at a steady 75 mph from 7:00am to 9:00am travels ∫[7 to 9] 75 dt = 75(9-7) = 150 miles. The integral represents the exact distance traveled.

Interestingly, functions with certain types of discontinuities can still be integrable. Consider f(x) = $x²-4$/$x-2$, which has a removable discontinuity at x=2 (a hole in the graph). Since this hole is infinitely small, it doesn't affect the area under the curve, so ∫[0 to 3] f(x)dx can still be calculated.

🧩 When integrating functions with discontinuities, it often helps to split the integral at the discontinuity points—but with removable discontinuities, the hole doesn't affect the overall area!

Using Technology for Definite Integrals

Your calculator is a powerful tool for evaluating complex definite integrals that would be difficult to solve by hand. The NINT (numerical integration) function lets you find approximate values for definite integrals quickly.

To use this function on your calculator, press MATH and select fnInt. Then enter the limits of integration, the function, and the independent variable. For example, you can find:

• ∫$-1 to 2$ x sin x dx ≈ 2.043
• ∫[0 to 2] 4/$1+x²$ dx ≈ 4.429
• ∫[0 to 5] e^$-x²$ dx ≈ 0.886

This calculation method is especially useful for integrals that don't have nice antiderivatives or involve complex functions.

💻 Remember that when using your calculator to find definite integrals, you should still write out the integral in your answer to show exactly what you're calculating!