Integral |sin(x)| from -pi/2 to pi/2, done fast in 2 minutes!
📚 Learn how to solve the definite integral of the absolute value of sine of x between -pi/2 and pi/2 in just 2 minutes with this fast tutorial! To solve the integral, we first have to break it into two separate pieces.
🔍 For the first piece, we evaluate the integral from -pi/2 to 0. Since the sine function is negative in this interval, we need to take its absolute value by multiplying it by -1, which gives us -sin(x). For the second piece, we evaluate the integral from 0 to pi/2, where the sine function is already positive. Here, we can simply use sin(x) as the integrand.
📝 To solve the integrals, we use the fundamental theorem of calculus, which states that if we integrate a function and then differentiate the result, we get the original function back. By applying this theorem, we find that the antiderivative of -sin(x) is cos(x), and the antiderivative of sin(x) is -cos(x).
📈 By plugging in the upper and lower limits, simplifying the expressions, and carefully evaluating the values, we arrive at the final answer of 2 for the definite integral of |sin(x)| from -pi/2 to pi/2.
🔑 Keywords definite integral, absolute value, sine function, fundamental theorem of calculus, antiderivative, cosine function.
❓ Common questions
1️⃣ What is the definite integral of |sin(x)| from -pi/2 to pi/2?
The definite integral of |sin(x)| from -pi/2 to pi/2 is 2.
2️⃣ How do you find the integral of absolute value functions?
To find the integral of absolute value functions, you need to break the function into pieces based on where it changes sign and evaluate each piece separately.
3️⃣ What is the fundamental theorem of calculus?
The fundamental theorem of calculus states that if we integrate a function and then differentiate the result, we get the original function back.
4️⃣ How do you integrate sin(x) and cos(x)?
The antiderivative of sin(x) is -cos(x), and the antiderivative of cos(x) is sin(x).
5️⃣ What is the antiderivative of -sin(x)?
The antiderivative of -sin(x) is cos(x).
6️⃣ How do you evaluate definite integrals?
To evaluate definite integrals, you need to plug in the upper and lower limits, simplify the expression, and carefully evaluate the values.
7️⃣ What is the unit circle?
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.
8️⃣ What is the cosine function?
The cosine function is a trigonometric function that gives the ratio of the adjacent side to the hypotenuse of a right triangle.
9️⃣ What is the sine function?
The sine function is a trigonometric function that gives the ratio of the opposite side to the hypotenuse of a right triangle.
🔟 What is the difference between definite and indefinite integrals?
A definite integral has limits of integration and gives a specific numerical value, while an indefinite integral has no limits of integration and gives a family of functions.
1️⃣1️⃣ What is the difference between sin(x) and -sin(x)?
The functions sin(x) and -sin(x) have the same shape but are reflected across the x-axis.
1️⃣2️⃣ What is the derivative of cos(x)?
The derivative of cos(x) is -sin(x).
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