### Home > APCALC > Chapter 3 > Lesson 3.4.3 > Problem3-177

3-177.

Write a Riemann sum to estimate the area under the curve for $0 ≤ x ≤ 2$ using $n$ left endpoint rectangles given $g(x) = −x^2 + 4$.

General left-endpoint Riemann sum:

$\displaystyle \sum_{n=0}^{n-1}\Delta xf(a+\Delta xi)$

$g(x)\approx \displaystyle \sum_{n=0}^{n-1}\frac{2}{n}\left ( a+\frac{2i}{n} \right )$

1. Calculate the sum for $n = 20$.

Substitute.

2. How can you use your result to estimate the area under the curve for $−2\le x\le0$? What about the area under the curve for $−2\le x\le2$?

Consider the shape of the graph of $g(x)$.
{"ft_lang_label":"__label__en","ft_lang_prob":0.74285394,"math_prob":0.9999585,"size":345,"snap":"2020-24-2020-29","text_gpt3_token_len":94,"char_repetition_ratio":0.16715543,"word_repetition_ratio":0.1904762,"special_character_ratio":0.27246377,"punctuation_ratio":0.12328767,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000002,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-03T17:31:23Z\",\"WARC-Record-ID\":\"<urn:uuid:f4e6d046-ce92-49ca-8a51-7dfee10f3794>\",\"Content-Length\":\"40527\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5ef4aa7c-1e88-423e-8200-32879941601a>\",\"WARC-Concurrent-To\":\"<urn:uuid:b493cbf5-df32-4f46-876f-cd55548ac2fb>\",\"WARC-IP-Address\":\"172.67.70.60\",\"WARC-Target-URI\":\"https://homework.cpm.org/category/CCI_CT/textbook/apcalc/chapter/3/lesson/3.4.3/problem/3-177\",\"WARC-Payload-Digest\":\"sha1:4EGS7FGIDLHFKL5ZNFAG33KLOBWM56XE\",\"WARC-Block-Digest\":\"sha1:37434TFPDXKAH2465DKEXN2G7EFC5SCD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655882634.5_warc_CC-MAIN-20200703153451-20200703183451-00461.warc.gz\"}"}